research article optimal robust motion controller design

Research ArticleOptimal Robust Motion Controller Design UsingMultiobjective Genetic Algorithm

Andrej Sarjaš1 Rajko SveIko1 and Amor Chowdhury12

1 Faculty of Electrical Engineering and Computer Science University of Maribor Smetanova 17 2000 Maribor Slovenia2Margento BV Crystal Tower Orlyplein 10 1043 DP Amsterdam The Netherlands

Correspondence should be addressed to Andrej Sarjas andrejsarjasumsi

Received 28 February 2014 Accepted 18 March 2014 Published 8 May 2014

Academic Editors N Barsoum V N Dieu P Vasant and G-W Weber

Copyright copy 2014 Andrej Sarjas et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper describes the use of a multiobjective genetic algorithm for robust motion controller design Motion controller structureis based on a disturbance observer in an RIC framework The RIC approach is presented in the form with internal and externalfeedback loops in which an internal disturbance rejection controller and an external performance controller must be synthesisedThis paper involves novel objectives for robustness and performance assessments for such an approach Objective functions forthe robustness property of RIC are based on simple even polynomials with nonnegativity conditions Regional pole placementmethod is presented with the aims of controllersrsquo structures simplification and their additional arbitrary selection Regional poleplacement involves arbitrary selection of central polynomials for both loops with additional admissible region of the optimizedpole location Polynomial deviation between selected and optimized polynomials is measured with derived performance objectivefunctions A multiobjective function is composed of different unrelated criteria such as robust stability controllersrsquo stability andtime-performance indexes of closed loopsThe design of controllers and multiobjective optimization procedure involve a set of theobjectives which are optimized simultaneously with a genetic algorithmmdashdifferential evolution

1 Introduction

Modern positioning systems require high performance andcomplex operation which demand highly efficient and com-plex controller structures Complex controller algorithmsand their designs are mostly related to very complex designprocedures and optimisation techniques with which design-ers try to achieve the desired performance specification Todate many advanced design methods have been developedbased on different structures of closed-loop systems andparadigmsThemost widely used designs are based on distur-bance observers [1ndash3] internal model control [4] and modelbased disturbance attenuation [5] The disturbance observerdesigns have been mostly used in industrial environmentsbecause of its simple transparent structure and capability toensure proper tradeoff between robustness and performanceproperties Tradeoffs between criteria can be handled withheuristic optimization approaches ormathematical program-ming optimization techniques abbreviated as LP QP SDPNP and so forth [6 7] Optimization-based methods in

computer aided design (CAD) have proved to be a valuabletool in engineering practice with which one can achieveproper performance and suitable controllers for closed-loopsystems In general the optimization procedure can bedivided into convex or nonconvex optimization problems

Mathematical programming approaches such as LP QPand SDP are very efficient for convex problems whichcover many modern robust controller designs in state spaceapproach [8ndash10] However there are also many controlproblems where the evaluation of the objective is not strictlyconvex or where the desired criteria cannot be combinedinto a set of convex objective functions For example poly-nomial synthesis with fixed order controller in a transferfunction form and controller structure optimization is abasically nonconvex problem [11 12] One way to over-come this problem is by combining heuristic optimizationmethods with new or conventional controller designs [13ndash15] This combination provides a set of efficient tools toaddress complex multivariable problems with performanceconstraints [13]There is much evidence which indicates high

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 978167 15 pageshttpdxdoiorg1011552014978167

2 The Scientific World Journal

rF(s)

e

minusC(s)

cPm(s)

Tin(s)K(s)

K(s)u i

din dout

P0(s) P998400(s)

120585998400

y

y

120585

minus

y

y

Figure 1 Disturbance observer in an RIC framework

efficiency of the heuristic optimization technique especiallywith genetic algorithms (GAs) The GA optimization tech-nique with well-formulated objective functions can preservesatisfactory results of controlled systems while overcomingsome problems and limitations of conventional designs [1316ndash18]

This paper considers a robust disturbance observer(DOB) in a robust internal compensator framework (RIC)with multiobjective optimization of different robustnessand performance criteria The DOB structure has alreadybeen studied in detail and has several different structuralapproaches [1 19] The more convenient and straightforwardmethods of DOB are based on Q-filter and inverse nominalmodels within the internal feedback branch [1 20 21] Themore advanced approaches are based on internal referencemodels in RIC framework [2 20] Bothmethods Q-filter andRIC deal well with disturbance attention and have similarapproaches and structures [2 20 22] However the RICapproach is more transparent and reliable in comparisonwith the Q-filter design The main difference between bothapproaches is in the design of their internal loops [5 22]Thesignificant disadvantage of the DOB with Q-filter design isthe usage of an inverse nominal model within the internalfeedback loop Most mechanical systems are presented asstrict proper functions which prevents direct usage of themodel inverse The second disadvantage is the internal-loopstructure with a low-pass Q-filter within a partially positivefeedback loopThe property of the Q-filter lowers the closed-looprsquos stability and the selection is strictly empirical withsome vague guidance for the first- second- and third-orderfilters [5 21 23 24] Each selection of approximated inversefunction and Q-filter requires additional assessment of theclosed-looprsquos performance and robustness The RIC on theother hand has better structural transparency and can bemuch easily incorporated into the optimization procedureInternal and external controllers can be acquired from theoptimization procedure so that the robustness of the RICsystem can be ensured [25 26]

This paper describes the design of an RIC disturbanceobserver with optimization of robustness and performancecriteria with multiobjective differential evolution (DE) [2728] The main contribution of the presented paper is amultiobjective optimization approach with simultaneousoptimization of a set of criteria for inner and outer loopsof the RIC structure The used robustness and performancecriteria are derived directly from the property of the norm119867infin

and uncertainty models [29 30] The criteria have theformof even polynomials where simple quasi-convex control

problems arise Robust stability of the system can be quitestraightforwardly assessed if the nonnegativity conditionof the polynomial is provided Even polynomials can bedirectly used in the optimization technique within geneticalgorithm without any other additional transformationThe second contribution of this paper is the design ofa simple parameterized controller structure with selectedcentral characteristic polynomial The central characteristicpolynomial has an admissible region in the stable half plainprescribed so that it ensures the expected performance andstability of the RIC system The basic idea comes fromthe pole-colouring technique and regional pole assignmentmethod [31 32] In comparison to similar methods thepresented approach uses objective functions of the regionalpole placement technique and can be used for nonparametricuncertainties The presented objectives are optimized with agenetic algorithm differential evolution (DE)DEhas evidentadvantages over other similar techniques simple structurefast convergence lower space complexity adaptive parametersettings and so forth [33] All design criteria within the DEalgorithm are evaluated over the nonnegativity property ofrobustness criteria and roots calculations of the closed-loopcharacteristic polynomial for regional pole placement

This paper is divided into seven sections In Section 2we describe the RIC disturbance observer for a positioningsystem Section 3 describes the regional pole placement forinternal and external loops of the RIC system and appliedparameterization of the controller structure Thereafter inSection 4 we describe performance and robustness proper-ties with the metric119867

infin based on nonnegativity assessments

of the even polynomial Section 5 describes a multiobjectiveoptimization approach with DE and a composite multiobjec-tive function After Section 5 a design example of an RICsystem with multiobjective optimization results is providedin Section 6The conclusions of the paper are summarized inSection 7

2 RIC Disturbance Observer forPositioning Systems

A disturbance observer in the RIC framework is composed ofinternal and external loops [34ndash36]Thedesign of the internalloop is based on input disturbance attention where mostoften appearing disturbances are reaction and load torquefriction and unmodeled dynamics [20 34]The external loopis designed so that it ensures the overall performance of theclosed-loop system The RIC structure is shown in Figure 1where119870(119904) 119875

0(119904) 1198751015840(119904) 119875

119898(119904) 119862(119904) and 119865(119904) are the internal

The Scientific World Journal 3

controller plant reference model and 2 DOF structureswith an external controller and a prefilter respectively Theinternal loopwith119875

0(119904) covers the angular velocity in relation

to input torques and disturbance rejection 119889in where 1198751015840

(119904)

is the transfer function of the rotary encoder The transferfunction of the rotary encoder is mostly treated as a systemwith pure integral behavior The RIC transfer functions are

1198750(119904) =

1198610(119904)

1198600(119904) 119875

1015840

(119904) =1198611015840

(119904)

1198601015840 (119904) 119870 (119904) =

119871119870(119904)

119877119870(119904)

119875119898(119904) =

119861119898(119904)

119860119898(119904) 119862 (119904) =

119871119862(119904)

119877119862(119904) 119865 (119904) =

119871119865(119904)

119877119865(119904)

(1)

The matrix form of the RIC system in Figure 1 is

[[[[[

[

119865minus1

0 0 119865minus1

0

minus119862 (119875119898119870 + 1) 1 0 0 119870

0 minus11987501198751015840

1 0 0

0 0 minus1 1 0

0 minus1198750

0 0 1

]]]]]

]

(

119890

119894

119910

119910

119910

)

=(

119903

119889in119889out120585

1205851015840

)

(2)

where 119903 119889in 119889out 120585 and 1205851015840 are reference signal input

disturbance output disturbance internal noise and externalnoise respectively

The closed-loop transfer function is

(

119890

119894

119910

119910

119910

) =1

1 + 1198701198750+ 119862119875

01198751015840 + 119870119862119875

11989811987501198751015840

times

[[[[[

[

119865 (1198701198750+ 1) minus119875

01198751015840

minus (1 + 1198701198750) minus (1 + 119870119875

0) 119875

01198751015840

119870

119865119862 (119875119898119870 + 1) 1 minus119862 (119875

119898119870 + 1) minus119862 (119875

119898119870 + 1) minus119870

11987501198751015840

119865119862 (119875119898119870 + 1) 119875

01198751015840

(1 + 1198701198750) minus119875

01198751015840

119862 (119875119898119870 + 1) minus119875

01198751015840

119865

11987501198751015840

119865119862 (119875119898119870 + 1) 119875

01198751015840

(1 + 1198701198750) (1 + 119870119875

0) minus119875

01198751015840

119865

1198750119865119862 (119875

119898119870 + 1) 119875

0minus119862119875

0(119875119898119870 + 1) minus119875

0119862 (119875

119898119870 + 1) (119875

01198751015840

119862 + 11987501198751015840

119862119875119898119870 + 1)

]]]]]

]

times (119903 119889in 119889out 120585 1205851015840

)119879

(3)

The RIC system in Figure 1 is slightly a modified clas-sic structure The internal loop in the modified structureembraces only the angular velocity while the classical internalloop uses for the feedback branch the same output as forthe external loop [21 22] The modified structure providesindirect decupling of the internal and external loops wherethe poles of the internal loop do not directly influencethe zeroes of the external loop Most RIC designs includestandard controller structures in internal loops [20 24]Standard controllers like PID or PI with integral behaviourimprove disturbance rejection for low-frequency disturbance119889in On the other hand internal integral behaviour causesa double-integrator effect on the external loop which sig-nificantly lowers the stability domain and generates unde-sired responses on the output disturbances and referencesignals This paper will consider an RIC structure shownin Figure 1 where the internal integral behaviour of thecontroller 119870(119904) does not directly influence the externalloop For the sake of the RIC structure simplification weassume that the prefilter function is constant and is equal to119865(119904) = 1

3 Regional Pole Placement of the RIC System

The design procedure of the RIC system can be addressed intwo steps the first step covers the internal loop controllerdesign and the second step the external loop controllerdesign Both controllers are designed with the regionalpole placement technique based on the selected centralpolynomial in the expected stable area The expected areais freely chosen by the designer and exhibits closed-loopdynamic performanceThe parameterization of controllers isapplied to ensure good disturbance rejection and robustnessproperties

Let us consider the internal loop of the system in Figure 1The closed-loop system is

(119910

119894) =

1

1 + 1198701198750

[1198750(1 + 119870119875

119898) 119875

0

1 + 119870119875119898

1](

119888

119889in)

(119910

119894) = 119878in [

1198750(1 + 119870119875

119898) 119875

0

1 + 119870119875119898

1](

119888

119889in)

(4)


Desired region of the poles

120590ts high

120590ts low

p desired locationp optimized location

j120596

p1

p3

p2

p2

p3

p1

120590

Figure 2 Settling-time objective function with settling-timeboundaries 120590

119905119904 low and 120590119905119904 high

where 119878in is the sensitivity function of the internal closed-loop The characteristic polynomial is defined as

1198600(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904) = 119862in (119904) (5)

Equation (5) is the standard starting point of pole place-ment design The solution of the equation is provided understrict polynomial degree conditions [37] The presentedapproach uses only the conditions when an exact solutiondoes not exist deg119877

119896le deg119860

0minus2 In this case the controller

structure does not depend on plant order like in the classicpole placement design and is arbitrarily selected by thedesigner The only way to satisfy expression (5) is regionalpole placementwith a prescribed region and deviation assess-ment as described in [32] Deviation is assessed betweenthe given polynomial 119862in(119904) and the candidate polynomialobtained with optimization The term prescribed region ofthe closed polynomial is similar to the term domain stability[29 38] The prescribed region with the selected centralpolynomial offers more leeway for further multiobjectiveoptimization of robustness and performance criteria Thepolynomial equation for regional pole placement is

1198600(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904) = 119862in (119904) 119862in (119904) asymp 119862in (119904)

(6)

where 119862in(119904) is the candidate polynomial obtained with theoptimization with controller coefficient 119870(119904) The prescribedregion of the desired pole location can be derived from theproperties of the Laplace stability domain119863in = 119904 = 120590+119895120596 isin

C | 120590 isin minusR119890 119895120596 isin I119898 where 119862in isin 119863in holds true Thepossible objective functions of the prescribed region for con-trollers119870(119904) 119862(119904) are described in the following subsections

31 Settling-Time Objective Function Roughly speaking thesettling time of the closed-loop system is inversely propor-tional to the real part of dominant poles in the complexdomain119863in The objective function can be composed sothat all closed-loop poles lie in a vertical stripe boundedby parameters120590

119905119904 high and 120590119905119904 low The vertical stripe belongs

to domain119863in Parameters120590119905119904 high and 120590

119905119904 low specify theobjective function criteria of the closed-loop settling time(see Figure 2)

The proposed objective function is

1198691= min

119870

(

1minussum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816minus1003816100381610038161003816120590119905119904 low

1003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816

forall120590119905119904 high lt R119890 119862in lt 120590119905119904 low and

sum

119899

10038161003816100381610038161003816

1003816100381610038161003816R119890 119862in1003816100381610038161003816 minus

10038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816

10038161003816100381610038161003816

max (1003816100381610038161003816R119890 119862in1003816100381610038161003816 R119890

10038161003816100381610038161003816119862in

10038161003816100381610038161003816)

) 119862in isin 119863in (7)

where 120590119905119904 high and 120590

119905119904 low indicate the location of the stripeboundary and 119899 indicates the number of closed-looppoles The right boundary 120590

119905119904 high of the stripe preventsthe possibility of obtaining poles with large negativereal part values Large negative real part pole values cancause a too high dynamic of the closed-loop system Ahigher dynamic of the closed-loop can render properreal-time implementation of the discrete controller on theembedded system or cause improper behaviour of controlleroutputs Improper higher dynamic of the controller outputmostly manifests itself as tempestuous responses whichindicate nonoptimal energy behaviour of the closed-loopsystem

The first term of the 1198691indicates a normalized value of

the objective if all poles lie on the left of the vertical line120590119905119904 lowThe second term assesses the deviation of the real part

pole value between the desired 119901 and the optimized 119901 polelocation where 119901 is 119901 isin 119862in119901 isin 119862in

32 Rise-TimeObjective Function Therise time of the closed-loop system is roughly proportional to the inverse of thedominant poles imaginary values andmay be upper-boundedby the parameter120596

119905119903Theobjective function can be composed

so that all closed-loop poles lie on the horizontally boundedstripe with boundsplusmn119895120596

119905119903(see Figure 3) The selected stripe

belongs to domain119863in


The proposed rise-time objective function is

1198692= min

119870

(

1minus sum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816minus1003816100381610038161003816120596119905119903

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816120596119905119903

1003816100381610038161003816

forall10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816lt1003816100381610038161003816120596119905119903

1003816100381610038161003816

sum

119899

10038161003816100381610038161003816

1003816100381610038161003816I119898119862in1003816100381610038161003816 minus

10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816

10038161003816100381610038161003816

max (1003816100381610038161003816I119898119862in1003816100381610038161003816 I119898

10038161003816100381610038161003816119862in

10038161003816100381610038161003816)

) 119862in isin 119863in (8)

The objective function 1198692describes the desired stripe in

the complex plain where the first term indicates the positioninside the horizontal stripe and the second the imaginarydeviation between the desired 119901 and the optimized 119901 poles

33 Damping-RatioObjective Function Damping ratio is alsoan important design criterion of the closed-loop dynamicperformance and is related to the angle of the vectors betweenthe negative real and the imaginary axis of the dominantpoles [32] The damping ratio is determined with the angleboundary plusmn120593

120577(see Figure 4)

The proposed damping-ratio objective function is

1198693= min

119870

(1 minussum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816120593 119862in

10038161003816100381610038161003816minus10038161003816100381610038161205931205891003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161205931205891003816100381610038161003816

forall10038161003816100381610038161003816120593 119862in

10038161003816100381610038161003816lt10038161003816100381610038161205931205891003816100381610038161003816 )

119862in isin 119863in

(9)

The proposed objective function 1198693represents the cone

region of the desired poles All objective functions 1198691ndash1198693

represent min-optimization procedure where all objectivesare normalized on the interval [0-1]

34 The Composite Objective Function of the Dynamic Crite-ria The composite objective function represents the inter-section of objectives 119869

1ndash1198693 A graphical presentation of the

composite objective function is shown in Figure 5The composite objective function can be presented as

multiobjective criteria for DE where the expected solutionarea is equal to

Ψin = (1198691 cap 1198692 cap 1198693) andmin (dev (119901in 119901in))

forallΨin isin 119863in

forall119901in = 119901in isin 119862in | 119862in isin 119863in


(10)

The multiobjective optimization approach will be dis-cussed in the following sections

35 Internal Controller Parameterization 119870(119904) Controllerparameterization is an applied technique which ensuressatisfied properties of the closed-loop system The internal

controllers 119870 and 119862 can be parameterized so as to ensuregood input disturbance rejection and robustness propertiesInmany applications the input disturbance 119889in in positioningsystems represents the load torque and Coulombrsquos frictionwith a typical low-frequency characteristic [20] The inter-nal system can provide good elimination of low-frequencydisturbances 119889in and reference tracking 119888 if the sensitivityfunction |119878in| ≪ 1 has a proper damping effect on thefrequency span 119861 = [120596

119897 120596ℎ] 119888(119861) and 119889in(119861) ≫ 1 Under

this assumption the controller119870 can be parameterized usingdifferent polynomial structures with a known effect on thesensitivity function |119878in| Controller parameterization withintegral action 119877

119870(119904) = 119877

1015840

119870(119904)119904 has a well-known influence

on the sensitivity function at low-frequency characteristicssuch as the known simple structures PI and PID For thesake of operational safety the integral action can in manycases be approximated with the stable pole (119904 + 120575) 120575 isin R |

0 lt 120575 ≪ 1 where 119870(119904) denominator polynomial is equalto 1198771015840

119870(119904)(119904 + 120575) In this case the damping of the sensitivity

function 119878in is lowered by parameter 120575 The lowered dampingvalue of 119878in is not noticeable in real-time operation especiallyif the absolute damping value is lower than the absolutemeasurement accuracy

A minimized tracking error 119890 and good low-frequencyinput disturbance rejection 119889in can be achieved if the follow-ing holds true

lim120596rarr119861

|119870 (119861)| ≫ 1 119861 = 119861 isin R | 0 lt 119861 le 120596low (11)

where 119861 is the low-frequency span of the sensitivity func-tion Sensitivity function 119878in for span 119861 with controllerparameterization (119904 + 120575) is

lim120596rarr119861

1003816100381610038161003816119878in (120575 120596)1003816100381610038161003816 = lim

120596rarr119861

(1003816100381610038161003816119878in (120596)

1003816100381610038161003816radic1205962 + 1205752)

lim120596rarr119861

|119878 (119861)| asymp 120575 997904rArr |119878 (120575 119861)| asymp 0

(12)

Parameter 120575 determines the closed-loop tracking accu-racy and the capability of disturbance rejection with charac-teristic polynomial119862in(119904) Controller structure 119870 is parame-terized as

119877119870(119904) =

1198771015840

119870(119904)

119901

prod

119896=1

(119904 + 120575119896)

1198771015840

119870(119904) 119877

119870par (119904)

(13)


Parameter 120575119896represents an additional parameter for

further optimization of performance and robustness criteriaThe polynomial equation with parametric solutions is

1198600(119904) 119877

1015840

119870(119904)

119901

prod

119896=1

(119904 + 120575119896) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

1198600(119904) 119877

119870(119904 120575) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

(14)

The solution of the optimization procedure is a properset of polynomial coefficients where the internal polynomial119862in(119904) belongs to the Ψin and a strong proximity condition119862in(119904) asymp 119862in(119904) holds true

36 The Design of the External Controller 119862(119904) Controller119862(119904)design technique is the same as for the internal controller119870(119904) where the external characteristic polynomial 119862out is

1198600(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119870(119904) 119877

119862(119904)

+ 1198610(119904) 119871

119870(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119862(119904)

+ 1198610(119904) 119861

1015840

(119904) 119871119862(119904) 119860

119898(119904) 119877

119870(119904)

+ 1198610(119904) 119861

1015840

(119904) 119861119898(119904) 119871

119870(119904) 119871

119862(119904) = 119862out (119904)

(15)

The possible controller parameterization 119862(119904) is

Γ (119904) 1198771015840

119862(119904)

119901

prod

119896=1

(119904 + 120578119896) + Υ (119904) 119871

119862(119904) = 119862out (119904)

Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871

119862(119904) = 119862out (119904)

(16)

where polynomials Γ(119904) and Υ(119904) are

Γ (119904) = 1198601015840

(119904) 119860119898(119904) (119860

0(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904))

Υ (119904) = 1198610(119904) 119861

1015840

(119904) (119860119898(119904) 119877

119870(119904) + 119861

119898(119904) 119871

119870(119904))

(17)

Coefficient 120578 can also be used as an additional parameterfor criteria optimization in the same sense as parameter 120575

The controller119862(119904) is the proper solution of the optimiza-tion problem for external loop if the following conditions aresatisfied

Ψout = (119869out 1 cap 119869out 2 cap 119869out 3) andmin (dev (119901out 119901out))

forallΨout isin 119863out

forall119901out = 119901out isin 119862out | 119862out isin 119863out

forall119901out = 119901out isin 119862in | 119862out isin 119863out

(18)

Objective functions 119869out 1ndash119869out 3 have the same propertiesand meanings as objective functions 119869

1ndash1198693

4 Nonparametric Uncertainty andRobustness Criterion of RIC

The robustness of the RIC system is assessed using uncer-tainty models Δ119875 and a stable proper input weight 1198811015840noise

120590

p1

p3

p2

p2

p3

p1

j120596

Frequency boundj120596tr

minusj120596tr

Figure 3 Rise-time objective function with a horizontal bound-aryplusmn119895120596

119905119903

120590

p1

p3

p2

p2

p3

p1

j120596Damping line

120593120577

minus120593120577

120577

120593p1

120593p2

Figure 4 Damping-ratio objective function with parameter plusmn120593120577

The uncertainty models describe model deviation within thegiven frequency space and are represented with a nominalmodel and stable uncertainty weights Δ119882 [30 39] Weights1198811015840

noise representmeasured noise spectrumof the signal11990810158401 For

RIC design we assume that the reference model is 119875119898asymp 119875

0isin

119875119867infin The internal loop of the RIC with uncertainty models

is shown in Figure 6The robustness assessment of the internal loop should

be considered with multiplicative and inverse uncertaintyThe multiplicative uncertainty represents uncertainties bylower frequencies while the inverse uncertainty representsuncertainties by higher frequencies [39]


σ

Stable area

Intersection

120590ts low120590ts high j120596

p2

p1

p3

120593120577

120577

j120596tr

minusj120596tr

Ψ

Figure 5 Intersection of objectives 1198691ndash1198693

c iPm(s) K(s 120590)

K(s 120590)

din

ΔP(s)y

120585998400w998400

1V998400noise(s)minus

Figure 6 Internal loop with uncertainty models Δ119875

Let us consider the multiplicative uncertainty modelΔ119875

119872= 119875

0(1 +Δ119882

119872)with nominal plant 119875

0 The closed-loop

characteristic using the multiplicative uncertainty modelΔ119875

119872is

(

119910

) = (1 + 119870119875

0(1 + Δ119882

119872))minus1

times [1198750(1 + Δ119882

119872) (1 + 119870119875

119898) 119875

0(1 + Δ119882

119872)

1 + 119870119875119898

1](

119888

119889in)

(19)

Robust stability for the multiplicative uncertainty is pre-served if the following holds true

10038171003817100381710038171003817100381710038171003817

Δ119882119872

1198701198750

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1

forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin

1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin

lt 1

(20)

where 119879in is a complementary sensitivity function of theinternal loop

+e c y

z

Ws(s)

C(s 120578) TP119898 in(s)y

P998400(s)

++minus

w1

dout

Vout(s)

120585Vnoise(s)

w2

Figure 7 External loop optimization structure

The internal loop with inverse uncertainty Δ119875119868

=

1198750(1 + Δ119882

119868)minus1 is defined as

(

119910

) = (1 + 119870119875

0(1 + Δ119882

119868)minus1

)minus1

times [1198750(1 + Δ119882

119868)minus1

(1 + 119870119875119898) 119875

0(1 + Δ119882

119868)minus1

1 + 119870119875119898

1]

times (119888

119889in)

(21)

The robustness for inverse models is satisfied if the followingholds true

10038171003817100381710038171003817100381710038171003817

Δ119882119868

1

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1

forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin (22)

where 119878in is the sensitivity function of the internal loopThe noise suppression 120585

1015840 of the internal loop can beassessed with the following criterion

1198791199081015840

1

= (1 + 119870 (120575) Δ119875)minus1

1198811015840

noise (23)

The robustness criterion of noise suppression with theuncertainty model is

1198791199081015840

1

= min119870

10038171003817100381710038171003817100381710038171003817

[119882119872

119882119868] [119879in 0

0 119878in]119881

1015840

noise

10038171003817100381710038171003817100381710038171003817

(24)

After derivation of robust stability conditions of theinternal loop robust stability conditions for the externalloop can be presented The purpose of external controllerdesign is to ensure overall performance characteristics likeproper reference tracking 119903 output disturbance rejection119889out and good measurement noise cancellation 120585 Theoptimization structure of the external loop is shown inFigure 7

Selected weights119882119904 119881out 119881noise belong to the PH

infin

domain Input weights119881out119881noise represent the input spectralcharacteristics of disturbance 119889out and noise 120585 where the per-formance weight119882

119904is selected to ensure a smooth frequency

characteristic of the external-loop sensitivity function 119878out(119904)A smooth characteristic of the sensitivity function is alsoan additional indicator of the closed-loop time-performancecharacteristics


The closed-loop characteristic of the external loop withweights119882

119904 119881out 119881noise is

119879119911119908= 119882

minus1

119904[(1 + 119862 (120578) 119879in119875

1015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

] [

[

1

119881out119881noise

]

]

= 119882minus1

119904[119878out 119878out 119878out] [

[

1


]

]

(25)

The optimal solution of the optimization problem is

min119862

10038171003817100381710038171198791199111199081003817100381710038171003817infin

= 120574min (26)

Themain goal of expression (26) is to minimize the influ-ences of inputs119908

1 119908

2on the sensitivity function 119878out and to

ensure final feedback performance with 119878out asymp 119882119904and |119878out| lt

|119882119904|

41 Robustness Criteria Assessment via Nonnegativity of theEven Polynomial Let us consider the property of the norm sdot

infinrelated to robust stability with uncertainty models

[30 39] The robust stability criterion is ensured withcondition sdot

infinlt 1 where the criterion is derived for a

given transfer function1198750(119904) = 119861(119904)119860

minus1

(119904) The condition1198750infin

lt 1 holds if the even polynomial 120587(1205962) is a strictlynonnegative function

119860(1205962

) minus 119861 (1205962

) = 120587 (1205962

) gt 0 (27)

The property of the function 120587(1205962) is derived from thefunctionΦ(119895120596) defined as in [30]

Φ(119895120596) = 1205742

119868 minus119861 (119895120596)

119860 (119895120596)

119861 (minus119895120596)

119860 (minus119895120596) (28)

Function Φ(119895120596) is a strictly continuous function for allisin R cup infin and has no imaginary zeroes The norm of thefunction equals 119875

0 le 120574 only if Φ(119895120596) gt 0 for all 120596 isin

R The robustness property for nonparametric uncertaintycan be assessed with even polynomial 120587(1205962) gt 0 (27)where we assume that 120574 = 1 and 119860(119895120596)119860(minus119895120596) = 119860(120596

2

)119861(119895120596)119861(minus119895120596) = 119861(120596

2

) holds true

Theorem 1 The norm sdot infin

of the system withpolynomials 119860 119861 is 119861119860

infinlt 1 only if the corresponding

polynomial120587(1205962) for all 120596 is a strictly nonnegative functionand 119861(119904)119860(119904)minus1 belongs to 119875119867

infin

Proof The simple explanation of the norm sdot infin

is119861119860

infin= sup

120596|119861(119895120596)119860

minus1

(119895120596)| where we assume that119861(119904)119860(119904)

minus1 belongs to 119875119867infin The norm of the transfer

function is 119861119860infin

lt 1 only if the ratio of polynomials|119861(119895120596)||119860(119895120596)|

minus1

lt 1 It is obvious from the above-mentioned that the difference (27) 120587(1205962) must be a strictly

nonnegative function and that 120587(1205962) has no real zeroes Thepositivity condition of the 120587(1205962) is an objective of robustnessassessment

Based on condition (27) the controllersrsquo robustnessproperty can be achieved with the assessment of the non-negativity of the even polynomial Each single robustnesscriterion ((20) (22) (24) and (26)) can be presentedin the form of an even polynomial and a nonnegativitycondition

42 Robust Stability Assessment with a NonnegativityCondition for the Internal Loop The internal-looprobustness conditions are presented with expressions(20) (22) and (24) Based on condition (27) the evenpolynomial 120587

119872(1205962

120575) for multiplicative uncertainty (20)is

120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908

119886119872(minus119904)

minus 1198610(119904) 119861

0(minus119904) 119871

119870(119904 120575) 119871

119870(minus119904 120575) 119908

119887119872(119904)

times 119908119887119872

(minus119904)

120587119872(1205962

120575) = 119862in119908119886119872 (1205962

) minus 1198610119871119870119908119887119872

(1205962

120575) gt 0

(29)

where the weight 119882119872(119904) is 119882

119872(119904) = 119908

119887119872(119904)119908

minus1

119886119872(119904)

Even polynomial 120587119868(1205962

120575) for robust stability withinverse uncertainty model (22) with stable weight 119882

119868(119904) =

119908119887119868(119904)119908

minus1

119886119868(119904) is

120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)

minus 1198600(119904) 119860

0(minus119904) 119877

119870(119904 120575) 119877

119870(minus119904 120575) 119908

119887119868(119904)

times 119908119887119868(minus119904)

120587119868(1205962

120575) = 119862in119908119886119868 (1205962

) minus 1198600119877119870119908119887119868(1205962

120575) gt 0

(30)


Even polynomial 1205871198791199081015840

1

(1205962

120575) for condition (24) withstability weights 119882

119872(119904) and 119882

119868(119904) and input weight

1198811015840

noise(119904) = V1015840119887noise(119904)V

1015840minus1

119886noise(119904) is

1205871198791199081015840

1119872

(1205962

120575) = 119908119886119872119862inV

1015840

119886noise (1205962

)

minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596

2

120575)

1205871198791199081015840

1119878

(1205962

120575) = 119908119886119878119862inV

1015840

119886out (1205962

)

minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596

2

120575)

1205871198791199081015840

1

(1205962

120575) = 05 (1205871198791199081015840

1119872

(1205962

120575) + 1205871198791199081015840

1119878

(1205962

120575)) gt 0

(31)

To simplify the optimization procedure in comparison tothe classic approach the composed condition (24) is treatedsimilarly as the criterion in augments plant transformationin a classic 119867

infindesign wherein the condition 120587

1198791199081015840

1

(1205962

120575)

(31) is a simple even polynomial function In the multi-objective optimization approach the condition 120587

1198791199081015840

1

(1205962

120575)

(31) can be also considered as two distinct functions1205871198791199081015840

1119872

(1205962

120575) and1205871198791199081015840

1119878

(1205962

120575)

43 Robust Stability Assessment with a Nonnegativity Con-dition for the External Loop A polynomial function forrobustness and performance condition (26) for the externalloop in Figure 7 can be formulated the sameway as conditions(24) and (31) The even polynomial of condition (26) withperformance weights 119882

119878(119904) = 119908

119887119878(119904)119908

minus1

119886119878(119904) 119881out(119904) =

V119887out(119904)V

minus1

119886out(119904) and 119881noise(119904) = V119887noise(119904)V

minus1

119886noise(119904) is

1205871199081119911(1205962

120578) = 119908119886119878119862out (120596

2

) minus 11990811988711987811986001198601198981198601015840

119877119870119877119862(1205962

120578)

1205871199082119911(1205962

120578) = 119908119886119878119862outV119886out (120596

2

)

minus 11990811988711987811986001198601198981198601015840

119877119870119877119862Vbout (120596

2

120578)

120587119911119908(1205962

120578) = (1205871199081119911(1205962

120578)2

+ 1205871199082119911(1205962

120578)2

) gt 0

(32)

As with condition (31) we can use composite objectivefunction 120587

119911119908(1205962

120578) to simplify the optimization procedureAdditional criteria are imposed to achieve strong stability

of the RIC where the real-time operational safety for thecontrolled systemmust be preserved in case some parts of thefeedback system fail for example sensorsrsquo failure electronicdriver failure and faulty motor Controllersrsquo stability canbe assessed over strict positive realness (SPR) criteria [40]Controllers119870(119904) 119862(119904) are stable transfer functions if SPRrsquosconditions hold true

R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0

R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)

where R119890[119870(119895120596)] and R119890[119862(119895120596)] are even functions Theeven polynomials of stability conditions (33) are

120587119870SPR

(1205962

120590)

= 2 (119877119870(minus119895120596 120590) 119871

119870(119895120596) + 119877

119870(119895120596 120590) 119871

119870(minus119895120596)) gt 0

120587119862SPR

(1205962

120578)

= 2 (119877119862(minus119895120596 120578) 119871

119862(119895120596) + 119877

119862(119895120596 120578) 119871

119862(minus119895120596)) gt 0

(34)

The robust conditions and the stability property of con-trollers (29)ndash(32) (34) can be directly used in amultiobjectiveoptimization algorithm with DE

5 Multiobjective Optimization withDE for RIC Design

The paper presents a multiobjective optimization procedurewith DE Genetic algorithm DE is known as a very efficientand powerful stochastic optimization procedure DE includessimilar operation steps as other GAs but in comparison withother classic GAs DE deviates the current population withscaled differences between two randomly selected members[28] There are many reasons for using the DE algorithm forsolving various optimization problems in different scientificdisciplines Compared to other similar algorithms DE has asimpler structure and is easier to implement on real problemsBecause of its simple structure it is suitable for large scaleoptimization problems The control parameters of DE havebeen well studied and their influence on the optimizationprocedure is well known

This paper presents the design of a robust efficientsimple and structured disturbance observer based on sim-ple objective functions for performance and robustnessassessment The main problem in optimal control designis formulating proper efficient objective functions whichcan be further used in a corresponding optimization algo-rithm Objective functions with their properties must ensureoptimal or suboptimal solutions of the given problem Thegiven optimization problem in RIC structure with regionalpole placement technique does not provide a convex set ofobjective functions There also do not exist general straight-forward procedures to convert such objectives to convexproblem which are mostly in conflict with each other andunrelated For example unrelated criteria are strong stabilityand robustness closed-loop pole location and controllerstability controller structure and robustness performanceand so forth In such cases where engineering simplificationsare needed it is very appropriate to use a metaheuristicoptimization tool like DE For this reason this paper doesnot deal with the efficiency of multiobjective DE algorithmsand their variants but only considers the efficiency of the usedapproach in an optimal control problem in anRIC frameworkwith formed objective functions for dynamic properties (7)ndash(9) and robustness (29)ndash(32) and (34)

Multiobjective optimization problems with DE involvemultiple objectives whichmust be optimized simultaneously


(1) Select central polynomials 119862in(119904) 119862119900ut(119904) and weights119882(119904) 119881(119904)(2) Select controllersrsquo structure 119870119862 and parameters 120590 120578

deg119870 = deg119862in minus deg1198750

deg119877119870= deg 119871

119870

deg119862 = deg119862out minus (deg119862in + deg119875119898+ deg1198751015840)

deg119877119862= deg 119871

119862

(3) Parametersrsquo selection of the DE-optimization algorithm(Number of parents-NP differential weight-F crossover probability-CR mutation strategy)

(4) while (min119891(1205962 119883))(5) DE selects value of parameters 120590 120578(6) Solve polynomial equations (14) (16)(7) Derive convex polynomials 120587(1205962 119883)(8) Find current Pareto-front of the 119891(1205962 119883)(9) end while

Algorithm 1 Multiobjective DE algorithm

and a set of possible solutions must be obtained The setof possible solutions is evaluated based on the concept ofdominance and Pareto optimality [14 17 33] The presentedRIC design uses DEMO algorithm presented by Robic andFilipic [42] The DEMO combines the advantages of DEwith the mechanisms of Pareto-based ranking and crowdingdistance sorting The advantage of the DEMO algorithm isthat it ensures convergence to the Pareto-front and a uniformspread of individuals along the front [42]

A multiobjective function is composed of derived objec-tive functions (7)ndash(9) (29)ndash(32) and (34) and is equal to

min1205962or119883

119891 (1205962

119883) =

[[[[[[[[[[[[[[[[

[

1198691(119883) and 119869out 1 (119883)

1198692(119883) and 119869out 2 (119883)

1198693(119883) and 119869out 3 (119883)

120587119872(1205962

119883)

120587119868(1205962

119883)

1205871198791199081015840

1

(1205962

119883)

120587119911119908(1205962

119883)

120587119870SPR

(1205962

119883)

120587119862SPR

(1205962

119883)

]]]]]]]]]]]]]]]]

]

st 119891 (1205962

119883) gt 0

119883 isin 119875

(35)

where 119875 is parameter space and 119883 is decisionvector Decision vector 119883 contains the coefficientsof polynomials 119871

119870 119877

119870 119871

119862 119877

119862(1) with possible

parameterization coefficients 120575 and 120578 Figure 8 showsthe structure of the decision variable119883

NoteThe solution119883 isin 119875 is Pareto-optimal if and only if theredoes not exist119883 isin 119875 that satisfies119891(1205962 119883) lt 119891(1205962 119883)

The optimization algorithm starts with the preselectedcentral polynomials 119862in119862out where the controllersrsquo degreesare prescribed in the same way as in the classic polynomialdesign (6) (16) [37]The degree of controller 119870 is equal to thecondition deg119870 = deg119862in minus deg119875

0 where deg119862in gt deg119875

0

holds true The degree of controller 119862 can be determined

Table 1 Parameters of an electromechanical system

Motor parameters119869 62 sdot 10

minus3 kgm2

119861 42 sdot 10minus3Nms

119896119875

0081NmA119877 79Ω119871 57mH

C

LK RK

K

LC RC

Figure 8 Structure of decision variable119883

the same way as for controller119870 where deg119862 = deg119862out minus

(deg119862in+deg119875119898+deg1198751015840

) and deg119862out ge (deg119862in+deg119875119898+deg1198751015840) holds true The optimization procedure is shown inAlgorithm 1

6 The Design Procedure for RIC Controllers

The proposed robust motion controller design is demon-strated by an electromechanical positioning system with theparameters given in Table 1

The parameters 119869 119861 119896119875 119877 and 119871 are rotor inertia

viscous friction torque factor terminal resistance and rotorinduction respectively To ensure simple controller struc-tures of 119870(119904) and 119862(119904) we use a simplified model of themechanical system 119875

0(119904) The simplification is often applica-

ble if the plant has more strongly expressed dominant stablepoles in comparison to other stable poles The given model is

1198750(119904) =

120596 (119904)

119894 (119904)=

119896119875

119869119904 + 119861=

1306

119904 + 0667

1198751015840

(119904) =120593 (119904)

120596 (119904)=1

119904

(36)


Table 2 Uncertainty parameters of the positioning system

Uncertainty parametersΔ119869 [24 divide 98] sdot 10

minus3 kgm2

Δ119861 [33 divide 65] sdot 10minus3Nms

Δ119896119875

[06 divide 13] NmA

Table 3 RIC control requirements

Feedback loop RIC requirementsSettling time 119905

119904lt 14 s

Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862

The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896

119890120596| lt |119880limit|

especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896

119890is an electromotive force

constantThe uncertainty plant is given by

Δ119875 (119904) =Δ119896

119875

Δ119869119904 + Δ119861 (37)

where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2

The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are

119882119872(119904) =

134 times 1199043

+ 1156 times 1199042

+ 032 times 119904 + 0062

1199043 + 157 times 1199042 + 048 times 119904 + 0096

119882119868(119904) =

065 times 1199042

+ 039 times 119904 + 0083

1199042 + 082 times 119904 + 017

1198811015840

noise (119904) =00023 times 119904 + 12 sdot 10

minus6

119904 + 9803

(38)

The desired requirements of the RIC systems are given inTable 3

The reference model119875119898(119904) is selected so as to ensure

proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875

0(119904) The selected 119875

119898(119904) is

119875119898(119904) =

2892

119904 + 15 (39)

Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain

Allowed poles region 119863in for 119862in

120590119905119904 low minus07

120590119905119904 high minus4

plusmn119895120596119905119903

plusmn11989510

plusmn120593120577

plusmn30∘

Table 5 Allowed region for optimized internal-loop poles 119862out(119904)

in a complex plain

Allowed poles region 119863out for 119862out

120590119905119904 low minus035

120590119905119904 high minus9

plusmn119895120596119905119903

plusmn11989514

plusmn120593120577

plusmn74∘

According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows

119882119878(119904) =

21 times 1199042

+ 046 times 119904 + 0001

1199042 + 398 times 119904 + 099

119881out (119904) =0002 times 119904 + 00012

119904 + 00014

119881noise (119904) =29 sdot 10

minus3

times 119904 + 12 sdot 10minus3

119904 + 1001 sdot 103

(40)

The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is

119870(119904 1199031198960) =

1198971198961119904 + 119897

1198960

1199031198961119904 + 120575

(41)

where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10

minus4

minus10minus2

]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour

Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875

0 The internal central

polynomial is

119862in (119904) = 1199042

+ 2 times 119904 + 1 (42)

The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4

The selected controller structure 119862(119904) is

119862 (119904 1199031198880) =

11989711988821199042

+ 1198971198881119904 + 119897

1198880

11990311988821199042 + 119903

1198881119904 + 120578

(43)


Table 6 Parameters of DE-optimization algorithm

Optimization algorithm parametersNumber of parents NP 50

Differential weight 119865 085Crossover probability CR 092

Mutation strategy DErand1bin

Table 7 Polynomials 119862in 119862in roots comparisons

Polynomials 119862in(119904) 119862in(119904)

minus1 minus0944 minus 119895004

minus1 minus0944 + 119895004

Table 8 Polynomials 119862out 119862out roots comparisons

Polynomials 119862out(119904) 119862out(119904)

minus34 minus 119895123 minus309 minus 119895102

minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048

The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875

119898(119904) the external central polynomial is

selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)

where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875

119898+ deg1198751015840) The selected allowed region119863out

of the optimized external-loop poles 119862out(119904) is presented inTable 5

The structure of the decision variable119883 for the givenexample is shown in Figure 9

The selected parameters of DE algorithm are presented inTable 6

7 Results

Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows

119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)

The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)

and the optimized polynomial 119862in(119904) is presented in Table 7

CK

lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578

Figure 9 The structure of the decision variable 119883 with 10 parame-ters

0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)

Reference signalWorst-case modelNominal model

minus4

minus2

0

2

4

Figure 10 Step-reference tracking

0 50 100 150 200 250 300Time (s)

Curr

ent (

A)

Worst-case modelNominal model

minus05

0

05

1

Figure 11 Output of the controller 119870(119904)

A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8

From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region

The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9

The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896

119875) chosen from

Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively



0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20

Figure 13 Output-voltage of the RIC system

The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17

Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879

1199081015840

1

infin

and 119879119908119911infin Table 9The

system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers

8 Conclusion

This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such

0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3

Input disturbance times01 Nm

Figure 14 Input-output disturbance rejection


0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03

Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion

Table 9 Value of robust criteria

Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047

as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867

infin1198672controller design Further work will be focused

on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number



0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6

Figure 17 Output voltage of the RIC system with disturbanceattention

of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003

[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009

[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997

[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International

Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005

[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001

[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997

[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004

[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008

[9] T Bakka and H R Karimi ldquoRobust 119867infin

dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012

[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867

infincontrolrdquo International Journal of Robust and Nonlinear

Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with

time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008

[12] K J Hunt ldquoPolynomial LQG and 119867infin

infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992

[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014

[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002

[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis

based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011



[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995

[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011

[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007

[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996

[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991


[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003

[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013

[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009

[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011

[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011

[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011

[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003

[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997

[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-

troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007

[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997

[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007

[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000

[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999

[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013

[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994

[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999

[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990


[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

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Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



rF(s)

e

minusC(s)

cPm(s)

Tin(s)K(s)

K(s)u i

din dout

P0(s) P998400(s)

120585998400

y

y

120585

minus

y

y

Figure 1 Disturbance observer in an RIC framework

efficiency of the heuristic optimization technique especiallywith genetic algorithms (GAs) The GA optimization tech-nique with well-formulated objective functions can preservesatisfactory results of controlled systems while overcomingsome problems and limitations of conventional designs [1316ndash18]

This paper considers a robust disturbance observer(DOB) in a robust internal compensator framework (RIC)with multiobjective optimization of different robustnessand performance criteria The DOB structure has alreadybeen studied in detail and has several different structuralapproaches [1 19] The more convenient and straightforwardmethods of DOB are based on Q-filter and inverse nominalmodels within the internal feedback branch [1 20 21] Themore advanced approaches are based on internal referencemodels in RIC framework [2 20] Bothmethods Q-filter andRIC deal well with disturbance attention and have similarapproaches and structures [2 20 22] However the RICapproach is more transparent and reliable in comparisonwith the Q-filter design The main difference between bothapproaches is in the design of their internal loops [5 22]Thesignificant disadvantage of the DOB with Q-filter design isthe usage of an inverse nominal model within the internalfeedback loop Most mechanical systems are presented asstrict proper functions which prevents direct usage of themodel inverse The second disadvantage is the internal-loopstructure with a low-pass Q-filter within a partially positivefeedback loopThe property of the Q-filter lowers the closed-looprsquos stability and the selection is strictly empirical withsome vague guidance for the first- second- and third-orderfilters [5 21 23 24] Each selection of approximated inversefunction and Q-filter requires additional assessment of theclosed-looprsquos performance and robustness The RIC on theother hand has better structural transparency and can bemuch easily incorporated into the optimization procedureInternal and external controllers can be acquired from theoptimization procedure so that the robustness of the RICsystem can be ensured [25 26]

This paper describes the design of an RIC disturbanceobserver with optimization of robustness and performancecriteria with multiobjective differential evolution (DE) [2728] The main contribution of the presented paper is amultiobjective optimization approach with simultaneousoptimization of a set of criteria for inner and outer loopsof the RIC structure The used robustness and performancecriteria are derived directly from the property of the norm119867infin

and uncertainty models [29 30] The criteria have theformof even polynomials where simple quasi-convex control

problems arise Robust stability of the system can be quitestraightforwardly assessed if the nonnegativity conditionof the polynomial is provided Even polynomials can bedirectly used in the optimization technique within geneticalgorithm without any other additional transformationThe second contribution of this paper is the design ofa simple parameterized controller structure with selectedcentral characteristic polynomial The central characteristicpolynomial has an admissible region in the stable half plainprescribed so that it ensures the expected performance andstability of the RIC system The basic idea comes fromthe pole-colouring technique and regional pole assignmentmethod [31 32] In comparison to similar methods thepresented approach uses objective functions of the regionalpole placement technique and can be used for nonparametricuncertainties The presented objectives are optimized with agenetic algorithm differential evolution (DE)DEhas evidentadvantages over other similar techniques simple structurefast convergence lower space complexity adaptive parametersettings and so forth [33] All design criteria within the DEalgorithm are evaluated over the nonnegativity property ofrobustness criteria and roots calculations of the closed-loopcharacteristic polynomial for regional pole placement

This paper is divided into seven sections In Section 2we describe the RIC disturbance observer for a positioningsystem Section 3 describes the regional pole placement forinternal and external loops of the RIC system and appliedparameterization of the controller structure Thereafter inSection 4 we describe performance and robustness proper-ties with the metric119867

infin based on nonnegativity assessments

of the even polynomial Section 5 describes a multiobjectiveoptimization approach with DE and a composite multiobjec-tive function After Section 5 a design example of an RICsystem with multiobjective optimization results is providedin Section 6The conclusions of the paper are summarized inSection 7

2 RIC Disturbance Observer forPositioning Systems

A disturbance observer in the RIC framework is composed ofinternal and external loops [34ndash36]Thedesign of the internalloop is based on input disturbance attention where mostoften appearing disturbances are reaction and load torquefriction and unmodeled dynamics [20 34]The external loopis designed so that it ensures the overall performance of theclosed-loop system The RIC structure is shown in Figure 1where119870(119904) 119875

0(119904) 1198751015840(119904) 119875

119898(119904) 119862(119904) and 119865(119904) are the internal





(119904)


1198750(119904) =

1198610(119904)

1198600(119904) 119875

1015840

(119904) =1198611015840

(119904)

1198601015840 (119904) 119870 (119904) =

119871119870(119904)

119877119870(119904)

119875119898(119904) =

119861119898(119904)

119860119898(119904) 119862 (119904) =

119871119862(119904)

119877119862(119904) 119865 (119904) =

119871119865(119904)

119877119865(119904)

(1)


[[[[[

[

119865minus1

0 0 119865minus1

0

minus119862 (119875119898119870 + 1) 1 0 0 119870

0 minus11987501198751015840

1 0 0

0 0 minus1 1 0

0 minus1198750

0 0 1

]]]]]

]

(

119890

119894

119910

119910

119910

)

=(

119903

119889in119889out120585

1205851015840

)

(2)




(

119890

119894

119910

119910

119910

) =1

1 + 1198701198750+ 119862119875

01198751015840 + 119870119862119875

11989811987501198751015840

times

[[[[[

[

119865 (1198701198750+ 1) minus119875

01198751015840

minus (1 + 1198701198750) minus (1 + 119870119875

0) 119875

01198751015840

119870

119865119862 (119875119898119870 + 1) 1 minus119862 (119875

119898119870 + 1) minus119862 (119875

119898119870 + 1) minus119870

11987501198751015840

119865119862 (119875119898119870 + 1) 119875

01198751015840

(1 + 1198701198750) minus119875

01198751015840

119862 (119875119898119870 + 1) minus119875

01198751015840

119865

11987501198751015840

119865119862 (119875119898119870 + 1) 119875

01198751015840

(1 + 1198701198750) (1 + 119870119875

0) minus119875

01198751015840

119865

1198750119865119862 (119875

119898119870 + 1) 119875

0minus119862119875

0(119875119898119870 + 1) minus119875

0119862 (119875

119898119870 + 1) (119875

01198751015840

119862 + 11987501198751015840

119862119875119898119870 + 1)

]]]]]

]

times (119903 119889in 119889out 120585 1205851015840

)119879

(3)





(119910

119894) =

1

1 + 1198701198750

[1198750(1 + 119870119875

119898) 119875

0

1 + 119870119875119898

1](

119888

119889in)

(119910

119894) = 119878in [

1198750(1 + 119870119875

119898) 119875

0

1 + 119870119875119898

1](

119888

119889in)

(4)



120590ts high

120590ts low


j120596

p1

p3

p2

p2

p3

p1

120590


119905119904 low and 120590119905119904 high


1198600(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904) = 119862in (119904) (5)


119896le deg119860



1198600(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904) = 119862in (119904) 119862in (119904) asymp 119862in (119904)

(6)








1198691= min

119870

(

1minussum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816minus1003816100381610038161003816120590119905119904 low

1003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816


sum

119899

10038161003816100381610038161003816

1003816100381610038161003816R119890 119862in1003816100381610038161003816 minus

10038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816

10038161003816100381610038161003816

max (1003816100381610038161003816R119890 119862in1003816100381610038161003816 R119890

10038161003816100381610038161003816119862in

10038161003816100381610038161003816)

) 119862in isin 119863in (7)

where 120590119905119904 high and 120590













1198692= min

119870

(

1minus sum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816minus1003816100381610038161003816120596119905119903

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816120596119905119903

1003816100381610038161003816

forall10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816lt1003816100381610038161003816120596119905119903

1003816100381610038161003816

sum

119899

10038161003816100381610038161003816

1003816100381610038161003816I119898119862in1003816100381610038161003816 minus

10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816

10038161003816100381610038161003816

max (1003816100381610038161003816I119898119862in1003816100381610038161003816 I119898

10038161003816100381610038161003816119862in

10038161003816100381610038161003816)

) 119862in isin 119863in (8)






1198693= min

119870

(1 minussum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816120593 119862in

10038161003816100381610038161003816minus10038161003816100381610038161205931205891003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161205931205891003816100381610038161003816

forall10038161003816100381610038161003816120593 119862in

10038161003816100381610038161003816lt10038161003816100381610038161205931205891003816100381610038161003816 )

119862in isin 119863in

(9)












(10)




119897 120596ℎ] 119888(119861) and 119889in(119861) ≫ 1 Under


119870(119904) = 119877

1015840







lim120596rarr119861

|119870 (119861)| ≫ 1 119861 = 119861 isin R | 0 lt 119861 le 120596low (11)


lim120596rarr119861

1003816100381610038161003816119878in (120575 120596)1003816100381610038161003816 = lim

120596rarr119861

(1003816100381610038161003816119878in (120596)

1003816100381610038161003816radic1205962 + 1205752)

lim120596rarr119861

|119878 (119861)| asymp 120575 997904rArr |119878 (120575 119861)| asymp 0

(12)


119877119870(119904) =

1198771015840

119870(119904)

119901

prod

119896=1

(119904 + 120575119896)

1198771015840

119870(119904) 119877

119870par (119904)

(13)




1198600(119904) 119877

1015840

119870(119904)

119901

prod

119896=1

(119904 + 120575119896) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

1198600(119904) 119877

119870(119904 120575) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

(14)



1198600(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119870(119904) 119877

119862(119904)

+ 1198610(119904) 119871

119870(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119862(119904)

+ 1198610(119904) 119861

1015840

(119904) 119871119862(119904) 119860

119898(119904) 119877

119870(119904)

+ 1198610(119904) 119861

1015840

(119904) 119861119898(119904) 119871

119870(119904) 119871

119862(119904) = 119862out (119904)

(15)


Γ (119904) 1198771015840

119862(119904)

119901

prod

119896=1

(119904 + 120578119896) + Υ (119904) 119871

119862(119904) = 119862out (119904)

Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871

119862(119904) = 119862out (119904)

(16)


Γ (119904) = 1198601015840

(119904) 119860119898(119904) (119860

0(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904))

Υ (119904) = 1198610(119904) 119861

1015840

(119904) (119860119898(119904) 119877

119870(119904) + 119861

119898(119904) 119871

119870(119904))

(17)







(18)


1ndash1198693



120590

p1

p3

p2

p2

p3

p1

j120596


minusj120596tr


119905119903

120590

p1

p3

p2

p2

p3

p1

j120596Damping line

120593120577

minus120593120577

120577

120593p1

120593p2





0isin





σ

Stable area

Intersection

120590ts low120590ts high j120596

p2

p1

p3

120593120577

120577

j120596tr

minusj120596tr

Ψ


c iPm(s) K(s 120590)

K(s 120590)

din

ΔP(s)y

120585998400w998400




119872= 119875

0(1 +Δ119882


0 The closed-loop


119872is

(

119910

) = (1 + 119870119875

0(1 + Δ119882

119872))minus1

times [1198750(1 + Δ119882

119872) (1 + 119870119875

119898) 119875

0(1 + Δ119882

119872)

1 + 119870119875119898

1](

119888

119889in)

(19)


10038171003817100381710038171003817100381710038171003817

Δ119882119872

1198701198750

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1


1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin

lt 1

(20)


+e c y

z

Ws(s)

C(s 120578) TP119898 in(s)y

P998400(s)

++minus

w1

dout

Vout(s)

120585Vnoise(s)

w2



=

1198750(1 + Δ119882


(

119910

) = (1 + 119870119875

0(1 + Δ119882

119868)minus1

)minus1

times [1198750(1 + Δ119882

119868)minus1

(1 + 119870119875119898) 119875

0(1 + Δ119882

119868)minus1

1 + 119870119875119898

1]

times (119888

119889in)

(21)


10038171003817100381710038171003817100381710038171003817

Δ119882119868

1

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1




1198791199081015840

1

= (1 + 119870 (120575) Δ119875)minus1

1198811015840

noise (23)


1198791199081015840

1

= min119870

10038171003817100381710038171003817100381710038171003817

[119882119872

119882119868] [119879in 0

0 119878in]119881

1015840

noise

10038171003817100381710038171003817100381710038171003817

(24)



infin






119904 119881out 119881noise is

119879119911119908= 119882

minus1

119904[(1 + 119862 (120578) 119879in119875

1015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

] [

[

1


]

]

= 119882minus1

119904[119878out 119878out 119878out] [

[

1


]

]

(25)


min119862

10038171003817100381710038171198791199111199081003817100381710038171003817infin

= 120574min (26)


1 119908



|119882119904|






minus1



119860(1205962

) minus 119861 (1205962

) = 120587 (1205962

) gt 0 (27)


Φ(119895120596) = 1205742

119868 minus119861 (119895120596)

119860 (119895120596)

119861 (minus119895120596)

119860 (minus119895120596) (28)




2

)119861(119895120596)119861(minus119895120596) = 119861(120596

2

) holds true





infin


is119861119860

infin= sup

120596|119861(119895120596)119860

minus1





minus1





119872(1205962


120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908

119886119872(minus119904)

minus 1198610(119904) 119861

0(minus119904) 119871

119870(119904 120575) 119871

119870(minus119904 120575) 119908

119887119872(119904)

times 119908119887119872

(minus119904)

120587119872(1205962

120575) = 119862in119908119886119872 (1205962

) minus 1198610119871119870119908119887119872

(1205962

120575) gt 0

(29)


119872(119904) = 119908

119887119872(119904)119908

minus1

119886119872(119904)

Even polynomial 120587119868(1205962


119868(119904) =

119908119887119868(119904)119908

minus1

119886119868(119904) is

120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)

minus 1198600(119904) 119860

0(minus119904) 119877

119870(119904 120575) 119877

119870(minus119904 120575) 119908

119887119868(119904)

times 119908119887119868(minus119904)

120587119868(1205962

120575) = 119862in119908119886119868 (1205962

) minus 1198600119877119870119908119887119868(1205962

120575) gt 0

(30)


Even polynomial 1205871198791199081015840

1

(1205962


119872(119904) and 119882


1198811015840

noise(119904) = V1015840119887noise(119904)V

1015840minus1

119886noise(119904) is

1205871198791199081015840

1119872

(1205962

120575) = 119908119886119872119862inV

1015840

119886noise (1205962

)

minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596

2

120575)

1205871198791199081015840

1119878

(1205962

120575) = 119908119886119878119862inV

1015840

119886out (1205962

)

minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596

2

120575)

1205871198791199081015840

1

(1205962

120575) = 05 (1205871198791199081015840

1119872

(1205962

120575) + 1205871198791199081015840

1119878

(1205962

120575)) gt 0

(31)



1198791199081015840

1

(1205962

120575)


1198791199081015840

1

(1205962

120575)


1119872

(1205962

120575) and1205871198791199081015840

1119878

(1205962

120575)


119878(119904) = 119908

119887119878(119904)119908

minus1

119886119878(119904) 119881out(119904) =

V119887out(119904)V

minus1


minus1

119886noise(119904) is

1205871199081119911(1205962

120578) = 119908119886119878119862out (120596

2

) minus 11990811988711987811986001198601198981198601015840

119877119870119877119862(1205962

120578)

1205871199082119911(1205962

120578) = 119908119886119878119862outV119886out (120596

2

)

minus 11990811988711987811986001198601198981198601015840

119877119870119877119862Vbout (120596

2

120578)

120587119911119908(1205962

120578) = (1205871199081119911(1205962

120578)2

+ 1205871199082119911(1205962

120578)2

) gt 0

(32)


119911119908(1205962



R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0

R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)


120587119870SPR

(1205962

120590)

= 2 (119877119870(minus119895120596 120590) 119871

119870(119895120596) + 119877

119870(119895120596 120590) 119871

119870(minus119895120596)) gt 0

120587119862SPR

(1205962

120578)

= 2 (119877119862(minus119895120596 120578) 119871

119862(119895120596) + 119877

119862(119895120596 120578) 119871

119862(minus119895120596)) gt 0

(34)









deg119877119870= deg 119871

119870


deg119877119862= deg 119871

119862






min1205962or119883

119891 (1205962

119883) =

[[[[[[[[[[[[[[[[

[

1198691(119883) and 119869out 1 (119883)

1198692(119883) and 119869out 2 (119883)

1198693(119883) and 119869out 3 (119883)

120587119872(1205962

119883)

120587119868(1205962

119883)

1205871198791199081015840

1

(1205962

119883)

120587119911119908(1205962

119883)

120587119870SPR

(1205962

119883)

120587119862SPR

(1205962

119883)

]]]]]]]]]]]]]]]]

]

st 119891 (1205962

119883) gt 0

119883 isin 119875

(35)


119870 119877

119870 119871

119862 119877






0




minus3 kgm2


119896119875

0081NmA119877 79Ω119871 57mH

C

LK RK

K

LC RC



(deg119862in+deg119875119898+deg1198751015840








1198750(119904) =

120596 (119904)

119894 (119904)=

119896119875

119869119904 + 119861=

1306

119904 + 0667

1198751015840

(119904) =120593 (119904)

120596 (119904)=1

119904

(36)




minus3 kgm2


Δ119896119875

[06 divide 13] NmA



119904lt 14 s



119890120596| lt |119880limit|




Δ119875 (119904) =Δ119896

119875

Δ119869119904 + Δ119861 (37)



119882119872(119904) =

134 times 1199043

+ 1156 times 1199042

+ 032 times 119904 + 0062

1199043 + 157 times 1199042 + 048 times 119904 + 0096

119882119868(119904) =

065 times 1199042

+ 039 times 119904 + 0083

1199042 + 082 times 119904 + 017

1198811015840

noise (119904) =00023 times 119904 + 12 sdot 10

minus6

119904 + 9803

(38)




0(119904) The selected 119875

119898(119904) is

119875119898(119904) =

2892

119904 + 15 (39)



120590119905119904 low minus07

120590119905119904 high minus4

plusmn119895120596119905119903

plusmn11989510

plusmn120593120577

plusmn30∘


in a complex plain


120590119905119904 low minus035

120590119905119904 high minus9

plusmn119895120596119905119903

plusmn11989514

plusmn120593120577

plusmn74∘


119882119878(119904) =

21 times 1199042

+ 046 times 119904 + 0001

1199042 + 398 times 119904 + 099

119881out (119904) =0002 times 119904 + 00012

119904 + 00014

119881noise (119904) =29 sdot 10

minus3


119904 + 1001 sdot 103

(40)


119870(119904 1199031198960) =

1198971198961119904 + 119897

1198960

1199031198961119904 + 120575

(41)


minus4

minus10minus2




polynomial is

119862in (119904) = 1199042

+ 2 times 119904 + 1 (42)



119862 (119904 1199031198880) =

11989711988821199042

+ 1198971198881119904 + 119897

1198880

11990311988821199042 + 119903

1198881119904 + 120578

(43)









minus1 minus0944 + 119895004




minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048



selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)






7 Results


119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)



CK



0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)


minus4

minus2

0

2

4


0 50 100 150 200 250 300Time (s)

Curr

ent (

A)


minus05

0

05

1






119875) chosen from




0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







(119904)


1198750(119904) =

1198610(119904)

1198600(119904) 119875

1015840

(119904) =1198611015840

(119904)

1198601015840 (119904) 119870 (119904) =

119871119870(119904)

119877119870(119904)

119875119898(119904) =

119861119898(119904)

119860119898(119904) 119862 (119904) =

119871119862(119904)

119877119862(119904) 119865 (119904) =

119871119865(119904)

119877119865(119904)

(1)


[[[[[

[

119865minus1

0 0 119865minus1

0

minus119862 (119875119898119870 + 1) 1 0 0 119870

0 minus11987501198751015840

1 0 0

0 0 minus1 1 0

0 minus1198750

0 0 1

]]]]]

]

(

119890

119894

119910

119910

119910

)

=(

119903

119889in119889out120585

1205851015840

)

(2)




(

119890

119894

119910

119910

119910

) =1

1 + 1198701198750+ 119862119875

01198751015840 + 119870119862119875

11989811987501198751015840

times

[[[[[

[

119865 (1198701198750+ 1) minus119875

01198751015840

minus (1 + 1198701198750) minus (1 + 119870119875

0) 119875

01198751015840

119870

119865119862 (119875119898119870 + 1) 1 minus119862 (119875

119898119870 + 1) minus119862 (119875

119898119870 + 1) minus119870

11987501198751015840

119865119862 (119875119898119870 + 1) 119875

01198751015840

(1 + 1198701198750) minus119875

01198751015840

119862 (119875119898119870 + 1) minus119875

01198751015840

119865

11987501198751015840

119865119862 (119875119898119870 + 1) 119875

01198751015840

(1 + 1198701198750) (1 + 119870119875

0) minus119875

01198751015840

119865

1198750119865119862 (119875

119898119870 + 1) 119875

0minus119862119875

0(119875119898119870 + 1) minus119875

0119862 (119875

119898119870 + 1) (119875

01198751015840

119862 + 11987501198751015840

119862119875119898119870 + 1)

]]]]]

]

times (119903 119889in 119889out 120585 1205851015840

)119879

(3)





(119910

119894) =

1

1 + 1198701198750

[1198750(1 + 119870119875

119898) 119875

0

1 + 119870119875119898

1](

119888

119889in)

(119910

119894) = 119878in [

1198750(1 + 119870119875

119898) 119875

0

1 + 119870119875119898

1](

119888

119889in)

(4)



120590ts high

120590ts low


j120596

p1

p3

p2

p2

p3

p1

120590


119905119904 low and 120590119905119904 high


1198600(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904) = 119862in (119904) (5)


119896le deg119860



1198600(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904) = 119862in (119904) 119862in (119904) asymp 119862in (119904)

(6)








1198691= min

119870

(

1minussum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816minus1003816100381610038161003816120590119905119904 low

1003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816


sum

119899

10038161003816100381610038161003816

1003816100381610038161003816R119890 119862in1003816100381610038161003816 minus

10038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816

10038161003816100381610038161003816

max (1003816100381610038161003816R119890 119862in1003816100381610038161003816 R119890

10038161003816100381610038161003816119862in

10038161003816100381610038161003816)

) 119862in isin 119863in (7)

where 120590119905119904 high and 120590













1198692= min

119870

(

1minus sum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816minus1003816100381610038161003816120596119905119903

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816120596119905119903

1003816100381610038161003816

forall10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816lt1003816100381610038161003816120596119905119903

1003816100381610038161003816

sum

119899

10038161003816100381610038161003816

1003816100381610038161003816I119898119862in1003816100381610038161003816 minus

10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816

10038161003816100381610038161003816

max (1003816100381610038161003816I119898119862in1003816100381610038161003816 I119898

10038161003816100381610038161003816119862in

10038161003816100381610038161003816)

) 119862in isin 119863in (8)






1198693= min

119870

(1 minussum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816120593 119862in

10038161003816100381610038161003816minus10038161003816100381610038161205931205891003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161205931205891003816100381610038161003816

forall10038161003816100381610038161003816120593 119862in

10038161003816100381610038161003816lt10038161003816100381610038161205931205891003816100381610038161003816 )

119862in isin 119863in

(9)












(10)




119897 120596ℎ] 119888(119861) and 119889in(119861) ≫ 1 Under


119870(119904) = 119877

1015840







lim120596rarr119861

|119870 (119861)| ≫ 1 119861 = 119861 isin R | 0 lt 119861 le 120596low (11)


lim120596rarr119861

1003816100381610038161003816119878in (120575 120596)1003816100381610038161003816 = lim

120596rarr119861

(1003816100381610038161003816119878in (120596)

1003816100381610038161003816radic1205962 + 1205752)

lim120596rarr119861

|119878 (119861)| asymp 120575 997904rArr |119878 (120575 119861)| asymp 0

(12)


119877119870(119904) =

1198771015840

119870(119904)

119901

prod

119896=1

(119904 + 120575119896)

1198771015840

119870(119904) 119877

119870par (119904)

(13)




1198600(119904) 119877

1015840

119870(119904)

119901

prod

119896=1

(119904 + 120575119896) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

1198600(119904) 119877

119870(119904 120575) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

(14)



1198600(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119870(119904) 119877

119862(119904)

+ 1198610(119904) 119871

119870(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119862(119904)

+ 1198610(119904) 119861

1015840

(119904) 119871119862(119904) 119860

119898(119904) 119877

119870(119904)

+ 1198610(119904) 119861

1015840

(119904) 119861119898(119904) 119871

119870(119904) 119871

119862(119904) = 119862out (119904)

(15)


Γ (119904) 1198771015840

119862(119904)

119901

prod

119896=1

(119904 + 120578119896) + Υ (119904) 119871

119862(119904) = 119862out (119904)

Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871

119862(119904) = 119862out (119904)

(16)


Γ (119904) = 1198601015840

(119904) 119860119898(119904) (119860

0(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904))

Υ (119904) = 1198610(119904) 119861

1015840

(119904) (119860119898(119904) 119877

119870(119904) + 119861

119898(119904) 119871

119870(119904))

(17)







(18)


1ndash1198693



120590

p1

p3

p2

p2

p3

p1

j120596


minusj120596tr


119905119903

120590

p1

p3

p2

p2

p3

p1

j120596Damping line

120593120577

minus120593120577

120577

120593p1

120593p2





0isin





σ

Stable area

Intersection

120590ts low120590ts high j120596

p2

p1

p3

120593120577

120577

j120596tr

minusj120596tr

Ψ


c iPm(s) K(s 120590)

K(s 120590)

din

ΔP(s)y

120585998400w998400




119872= 119875

0(1 +Δ119882


0 The closed-loop


119872is

(

119910

) = (1 + 119870119875

0(1 + Δ119882

119872))minus1

times [1198750(1 + Δ119882

119872) (1 + 119870119875

119898) 119875

0(1 + Δ119882

119872)

1 + 119870119875119898

1](

119888

119889in)

(19)


10038171003817100381710038171003817100381710038171003817

Δ119882119872

1198701198750

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1


1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin

lt 1

(20)


+e c y

z

Ws(s)

C(s 120578) TP119898 in(s)y

P998400(s)

++minus

w1

dout

Vout(s)

120585Vnoise(s)

w2



=

1198750(1 + Δ119882


(

119910

) = (1 + 119870119875

0(1 + Δ119882

119868)minus1

)minus1

times [1198750(1 + Δ119882

119868)minus1

(1 + 119870119875119898) 119875

0(1 + Δ119882

119868)minus1

1 + 119870119875119898

1]

times (119888

119889in)

(21)


10038171003817100381710038171003817100381710038171003817

Δ119882119868

1

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1




1198791199081015840

1

= (1 + 119870 (120575) Δ119875)minus1

1198811015840

noise (23)


1198791199081015840

1

= min119870

10038171003817100381710038171003817100381710038171003817

[119882119872

119882119868] [119879in 0

0 119878in]119881

1015840

noise

10038171003817100381710038171003817100381710038171003817

(24)



infin






119904 119881out 119881noise is

119879119911119908= 119882

minus1

119904[(1 + 119862 (120578) 119879in119875

1015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

] [

[

1


]

]

= 119882minus1

119904[119878out 119878out 119878out] [

[

1


]

]

(25)


min119862

10038171003817100381710038171198791199111199081003817100381710038171003817infin

= 120574min (26)


1 119908



|119882119904|






minus1



119860(1205962

) minus 119861 (1205962

) = 120587 (1205962

) gt 0 (27)


Φ(119895120596) = 1205742

119868 minus119861 (119895120596)

119860 (119895120596)

119861 (minus119895120596)

119860 (minus119895120596) (28)




2

)119861(119895120596)119861(minus119895120596) = 119861(120596

2

) holds true





infin


is119861119860

infin= sup

120596|119861(119895120596)119860

minus1





minus1





119872(1205962


120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908

119886119872(minus119904)

minus 1198610(119904) 119861

0(minus119904) 119871

119870(119904 120575) 119871

119870(minus119904 120575) 119908

119887119872(119904)

times 119908119887119872

(minus119904)

120587119872(1205962

120575) = 119862in119908119886119872 (1205962

) minus 1198610119871119870119908119887119872

(1205962

120575) gt 0

(29)


119872(119904) = 119908

119887119872(119904)119908

minus1

119886119872(119904)

Even polynomial 120587119868(1205962


119868(119904) =

119908119887119868(119904)119908

minus1

119886119868(119904) is

120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)

minus 1198600(119904) 119860

0(minus119904) 119877

119870(119904 120575) 119877

119870(minus119904 120575) 119908

119887119868(119904)

times 119908119887119868(minus119904)

120587119868(1205962

120575) = 119862in119908119886119868 (1205962

) minus 1198600119877119870119908119887119868(1205962

120575) gt 0

(30)


Even polynomial 1205871198791199081015840

1

(1205962


119872(119904) and 119882


1198811015840

noise(119904) = V1015840119887noise(119904)V

1015840minus1

119886noise(119904) is

1205871198791199081015840

1119872

(1205962

120575) = 119908119886119872119862inV

1015840

119886noise (1205962

)

minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596

2

120575)

1205871198791199081015840

1119878

(1205962

120575) = 119908119886119878119862inV

1015840

119886out (1205962

)

minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596

2

120575)

1205871198791199081015840

1

(1205962

120575) = 05 (1205871198791199081015840

1119872

(1205962

120575) + 1205871198791199081015840

1119878

(1205962

120575)) gt 0

(31)



1198791199081015840

1

(1205962

120575)


1198791199081015840

1

(1205962

120575)


1119872

(1205962

120575) and1205871198791199081015840

1119878

(1205962

120575)


119878(119904) = 119908

119887119878(119904)119908

minus1

119886119878(119904) 119881out(119904) =

V119887out(119904)V

minus1


minus1

119886noise(119904) is

1205871199081119911(1205962

120578) = 119908119886119878119862out (120596

2

) minus 11990811988711987811986001198601198981198601015840

119877119870119877119862(1205962

120578)

1205871199082119911(1205962

120578) = 119908119886119878119862outV119886out (120596

2

)

minus 11990811988711987811986001198601198981198601015840

119877119870119877119862Vbout (120596

2

120578)

120587119911119908(1205962

120578) = (1205871199081119911(1205962

120578)2

+ 1205871199082119911(1205962

120578)2

) gt 0

(32)


119911119908(1205962



R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0

R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)


120587119870SPR

(1205962

120590)

= 2 (119877119870(minus119895120596 120590) 119871

119870(119895120596) + 119877

119870(119895120596 120590) 119871

119870(minus119895120596)) gt 0

120587119862SPR

(1205962

120578)

= 2 (119877119862(minus119895120596 120578) 119871

119862(119895120596) + 119877

119862(119895120596 120578) 119871

119862(minus119895120596)) gt 0

(34)









deg119877119870= deg 119871

119870


deg119877119862= deg 119871

119862






min1205962or119883

119891 (1205962

119883) =

[[[[[[[[[[[[[[[[

[

1198691(119883) and 119869out 1 (119883)

1198692(119883) and 119869out 2 (119883)

1198693(119883) and 119869out 3 (119883)

120587119872(1205962

119883)

120587119868(1205962

119883)

1205871198791199081015840

1

(1205962

119883)

120587119911119908(1205962

119883)

120587119870SPR

(1205962

119883)

120587119862SPR

(1205962

119883)

]]]]]]]]]]]]]]]]

]

st 119891 (1205962

119883) gt 0

119883 isin 119875

(35)


119870 119877

119870 119871

119862 119877






0




minus3 kgm2


119896119875

0081NmA119877 79Ω119871 57mH

C

LK RK

K

LC RC



(deg119862in+deg119875119898+deg1198751015840








1198750(119904) =

120596 (119904)

119894 (119904)=

119896119875

119869119904 + 119861=

1306

119904 + 0667

1198751015840

(119904) =120593 (119904)

120596 (119904)=1

119904

(36)




minus3 kgm2


Δ119896119875

[06 divide 13] NmA



119904lt 14 s



119890120596| lt |119880limit|




Δ119875 (119904) =Δ119896

119875

Δ119869119904 + Δ119861 (37)



119882119872(119904) =

134 times 1199043

+ 1156 times 1199042

+ 032 times 119904 + 0062

1199043 + 157 times 1199042 + 048 times 119904 + 0096

119882119868(119904) =

065 times 1199042

+ 039 times 119904 + 0083

1199042 + 082 times 119904 + 017

1198811015840

noise (119904) =00023 times 119904 + 12 sdot 10

minus6

119904 + 9803

(38)




0(119904) The selected 119875

119898(119904) is

119875119898(119904) =

2892

119904 + 15 (39)



120590119905119904 low minus07

120590119905119904 high minus4

plusmn119895120596119905119903

plusmn11989510

plusmn120593120577

plusmn30∘


in a complex plain


120590119905119904 low minus035

120590119905119904 high minus9

plusmn119895120596119905119903

plusmn11989514

plusmn120593120577

plusmn74∘


119882119878(119904) =

21 times 1199042

+ 046 times 119904 + 0001

1199042 + 398 times 119904 + 099

119881out (119904) =0002 times 119904 + 00012

119904 + 00014

119881noise (119904) =29 sdot 10

minus3


119904 + 1001 sdot 103

(40)


119870(119904 1199031198960) =

1198971198961119904 + 119897

1198960

1199031198961119904 + 120575

(41)


minus4

minus10minus2




polynomial is

119862in (119904) = 1199042

+ 2 times 119904 + 1 (42)



119862 (119904 1199031198880) =

11989711988821199042

+ 1198971198881119904 + 119897

1198880

11990311988821199042 + 119903

1198881119904 + 120578

(43)









minus1 minus0944 + 119895004




minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048



selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)






7 Results


119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)



CK



0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)


minus4

minus2

0

2

4


0 50 100 150 200 250 300Time (s)

Curr

ent (

A)


minus05

0

05

1






119875) chosen from




0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





120590ts high

120590ts low


j120596

p1

p3

p2

p2

p3

p1

120590


119905119904 low and 120590119905119904 high


1198600(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904) = 119862in (119904) (5)


119896le deg119860



1198600(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904) = 119862in (119904) 119862in (119904) asymp 119862in (119904)

(6)








1198691= min

119870

(

1minussum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816minus1003816100381610038161003816120590119905119904 low

1003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816


sum

119899

10038161003816100381610038161003816

1003816100381610038161003816R119890 119862in1003816100381610038161003816 minus

10038161003816100381610038161003816R119890 119862in

10038161003816100381610038161003816

10038161003816100381610038161003816

max (1003816100381610038161003816R119890 119862in1003816100381610038161003816 R119890

10038161003816100381610038161003816119862in

10038161003816100381610038161003816)

) 119862in isin 119863in (7)

where 120590119905119904 high and 120590













1198692= min

119870

(

1minus sum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816minus1003816100381610038161003816120596119905119903

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816120596119905119903

1003816100381610038161003816

forall10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816lt1003816100381610038161003816120596119905119903

1003816100381610038161003816

sum

119899

10038161003816100381610038161003816

1003816100381610038161003816I119898119862in1003816100381610038161003816 minus

10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816

10038161003816100381610038161003816

max (1003816100381610038161003816I119898119862in1003816100381610038161003816 I119898

10038161003816100381610038161003816119862in

10038161003816100381610038161003816)

) 119862in isin 119863in (8)






1198693= min

119870

(1 minussum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816120593 119862in

10038161003816100381610038161003816minus10038161003816100381610038161205931205891003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161205931205891003816100381610038161003816

forall10038161003816100381610038161003816120593 119862in

10038161003816100381610038161003816lt10038161003816100381610038161205931205891003816100381610038161003816 )

119862in isin 119863in

(9)












(10)




119897 120596ℎ] 119888(119861) and 119889in(119861) ≫ 1 Under


119870(119904) = 119877

1015840







lim120596rarr119861

|119870 (119861)| ≫ 1 119861 = 119861 isin R | 0 lt 119861 le 120596low (11)


lim120596rarr119861

1003816100381610038161003816119878in (120575 120596)1003816100381610038161003816 = lim

120596rarr119861

(1003816100381610038161003816119878in (120596)

1003816100381610038161003816radic1205962 + 1205752)

lim120596rarr119861

|119878 (119861)| asymp 120575 997904rArr |119878 (120575 119861)| asymp 0

(12)


119877119870(119904) =

1198771015840

119870(119904)

119901

prod

119896=1

(119904 + 120575119896)

1198771015840

119870(119904) 119877

119870par (119904)

(13)




1198600(119904) 119877

1015840

119870(119904)

119901

prod

119896=1

(119904 + 120575119896) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

1198600(119904) 119877

119870(119904 120575) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

(14)



1198600(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119870(119904) 119877

119862(119904)

+ 1198610(119904) 119871

119870(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119862(119904)

+ 1198610(119904) 119861

1015840

(119904) 119871119862(119904) 119860

119898(119904) 119877

119870(119904)

+ 1198610(119904) 119861

1015840

(119904) 119861119898(119904) 119871

119870(119904) 119871

119862(119904) = 119862out (119904)

(15)


Γ (119904) 1198771015840

119862(119904)

119901

prod

119896=1

(119904 + 120578119896) + Υ (119904) 119871

119862(119904) = 119862out (119904)

Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871

119862(119904) = 119862out (119904)

(16)


Γ (119904) = 1198601015840

(119904) 119860119898(119904) (119860

0(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904))

Υ (119904) = 1198610(119904) 119861

1015840

(119904) (119860119898(119904) 119877

119870(119904) + 119861

119898(119904) 119871

119870(119904))

(17)







(18)


1ndash1198693



120590

p1

p3

p2

p2

p3

p1

j120596


minusj120596tr


119905119903

120590

p1

p3

p2

p2

p3

p1

j120596Damping line

120593120577

minus120593120577

120577

120593p1

120593p2





0isin





σ

Stable area

Intersection

120590ts low120590ts high j120596

p2

p1

p3

120593120577

120577

j120596tr

minusj120596tr

Ψ


c iPm(s) K(s 120590)

K(s 120590)

din

ΔP(s)y

120585998400w998400




119872= 119875

0(1 +Δ119882


0 The closed-loop


119872is

(

119910

) = (1 + 119870119875

0(1 + Δ119882

119872))minus1

times [1198750(1 + Δ119882

119872) (1 + 119870119875

119898) 119875

0(1 + Δ119882

119872)

1 + 119870119875119898

1](

119888

119889in)

(19)


10038171003817100381710038171003817100381710038171003817

Δ119882119872

1198701198750

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1


1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin

lt 1

(20)


+e c y

z

Ws(s)

C(s 120578) TP119898 in(s)y

P998400(s)

++minus

w1

dout

Vout(s)

120585Vnoise(s)

w2



=

1198750(1 + Δ119882


(

119910

) = (1 + 119870119875

0(1 + Δ119882

119868)minus1

)minus1

times [1198750(1 + Δ119882

119868)minus1

(1 + 119870119875119898) 119875

0(1 + Δ119882

119868)minus1

1 + 119870119875119898

1]

times (119888

119889in)

(21)


10038171003817100381710038171003817100381710038171003817

Δ119882119868

1

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1




1198791199081015840

1

= (1 + 119870 (120575) Δ119875)minus1

1198811015840

noise (23)


1198791199081015840

1

= min119870

10038171003817100381710038171003817100381710038171003817

[119882119872

119882119868] [119879in 0

0 119878in]119881

1015840

noise

10038171003817100381710038171003817100381710038171003817

(24)



infin






119904 119881out 119881noise is

119879119911119908= 119882

minus1

119904[(1 + 119862 (120578) 119879in119875

1015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

] [

[

1


]

]

= 119882minus1

119904[119878out 119878out 119878out] [

[

1


]

]

(25)


min119862

10038171003817100381710038171198791199111199081003817100381710038171003817infin

= 120574min (26)


1 119908



|119882119904|






minus1



119860(1205962

) minus 119861 (1205962

) = 120587 (1205962

) gt 0 (27)


Φ(119895120596) = 1205742

119868 minus119861 (119895120596)

119860 (119895120596)

119861 (minus119895120596)

119860 (minus119895120596) (28)




2

)119861(119895120596)119861(minus119895120596) = 119861(120596

2

) holds true





infin


is119861119860

infin= sup

120596|119861(119895120596)119860

minus1





minus1





119872(1205962


120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908

119886119872(minus119904)

minus 1198610(119904) 119861

0(minus119904) 119871

119870(119904 120575) 119871

119870(minus119904 120575) 119908

119887119872(119904)

times 119908119887119872

(minus119904)

120587119872(1205962

120575) = 119862in119908119886119872 (1205962

) minus 1198610119871119870119908119887119872

(1205962

120575) gt 0

(29)


119872(119904) = 119908

119887119872(119904)119908

minus1

119886119872(119904)

Even polynomial 120587119868(1205962


119868(119904) =

119908119887119868(119904)119908

minus1

119886119868(119904) is

120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)

minus 1198600(119904) 119860

0(minus119904) 119877

119870(119904 120575) 119877

119870(minus119904 120575) 119908

119887119868(119904)

times 119908119887119868(minus119904)

120587119868(1205962

120575) = 119862in119908119886119868 (1205962

) minus 1198600119877119870119908119887119868(1205962

120575) gt 0

(30)


Even polynomial 1205871198791199081015840

1

(1205962


119872(119904) and 119882


1198811015840

noise(119904) = V1015840119887noise(119904)V

1015840minus1

119886noise(119904) is

1205871198791199081015840

1119872

(1205962

120575) = 119908119886119872119862inV

1015840

119886noise (1205962

)

minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596

2

120575)

1205871198791199081015840

1119878

(1205962

120575) = 119908119886119878119862inV

1015840

119886out (1205962

)

minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596

2

120575)

1205871198791199081015840

1

(1205962

120575) = 05 (1205871198791199081015840

1119872

(1205962

120575) + 1205871198791199081015840

1119878

(1205962

120575)) gt 0

(31)



1198791199081015840

1

(1205962

120575)


1198791199081015840

1

(1205962

120575)


1119872

(1205962

120575) and1205871198791199081015840

1119878

(1205962

120575)


119878(119904) = 119908

119887119878(119904)119908

minus1

119886119878(119904) 119881out(119904) =

V119887out(119904)V

minus1


minus1

119886noise(119904) is

1205871199081119911(1205962

120578) = 119908119886119878119862out (120596

2

) minus 11990811988711987811986001198601198981198601015840

119877119870119877119862(1205962

120578)

1205871199082119911(1205962

120578) = 119908119886119878119862outV119886out (120596

2

)

minus 11990811988711987811986001198601198981198601015840

119877119870119877119862Vbout (120596

2

120578)

120587119911119908(1205962

120578) = (1205871199081119911(1205962

120578)2

+ 1205871199082119911(1205962

120578)2

) gt 0

(32)


119911119908(1205962



R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0

R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)


120587119870SPR

(1205962

120590)

= 2 (119877119870(minus119895120596 120590) 119871

119870(119895120596) + 119877

119870(119895120596 120590) 119871

119870(minus119895120596)) gt 0

120587119862SPR

(1205962

120578)

= 2 (119877119862(minus119895120596 120578) 119871

119862(119895120596) + 119877

119862(119895120596 120578) 119871

119862(minus119895120596)) gt 0

(34)









deg119877119870= deg 119871

119870


deg119877119862= deg 119871

119862






min1205962or119883

119891 (1205962

119883) =

[[[[[[[[[[[[[[[[

[

1198691(119883) and 119869out 1 (119883)

1198692(119883) and 119869out 2 (119883)

1198693(119883) and 119869out 3 (119883)

120587119872(1205962

119883)

120587119868(1205962

119883)

1205871198791199081015840

1

(1205962

119883)

120587119911119908(1205962

119883)

120587119870SPR

(1205962

119883)

120587119862SPR

(1205962

119883)

]]]]]]]]]]]]]]]]

]

st 119891 (1205962

119883) gt 0

119883 isin 119875

(35)


119870 119877

119870 119871

119862 119877






0




minus3 kgm2


119896119875

0081NmA119877 79Ω119871 57mH

C

LK RK

K

LC RC



(deg119862in+deg119875119898+deg1198751015840








1198750(119904) =

120596 (119904)

119894 (119904)=

119896119875

119869119904 + 119861=

1306

119904 + 0667

1198751015840

(119904) =120593 (119904)

120596 (119904)=1

119904

(36)




minus3 kgm2


Δ119896119875

[06 divide 13] NmA



119904lt 14 s



119890120596| lt |119880limit|




Δ119875 (119904) =Δ119896

119875

Δ119869119904 + Δ119861 (37)



119882119872(119904) =

134 times 1199043

+ 1156 times 1199042

+ 032 times 119904 + 0062

1199043 + 157 times 1199042 + 048 times 119904 + 0096

119882119868(119904) =

065 times 1199042

+ 039 times 119904 + 0083

1199042 + 082 times 119904 + 017

1198811015840

noise (119904) =00023 times 119904 + 12 sdot 10

minus6

119904 + 9803

(38)




0(119904) The selected 119875

119898(119904) is

119875119898(119904) =

2892

119904 + 15 (39)



120590119905119904 low minus07

120590119905119904 high minus4

plusmn119895120596119905119903

plusmn11989510

plusmn120593120577

plusmn30∘


in a complex plain


120590119905119904 low minus035

120590119905119904 high minus9

plusmn119895120596119905119903

plusmn11989514

plusmn120593120577

plusmn74∘


119882119878(119904) =

21 times 1199042

+ 046 times 119904 + 0001

1199042 + 398 times 119904 + 099

119881out (119904) =0002 times 119904 + 00012

119904 + 00014

119881noise (119904) =29 sdot 10

minus3


119904 + 1001 sdot 103

(40)


119870(119904 1199031198960) =

1198971198961119904 + 119897

1198960

1199031198961119904 + 120575

(41)


minus4

minus10minus2




polynomial is

119862in (119904) = 1199042

+ 2 times 119904 + 1 (42)



119862 (119904 1199031198880) =

11989711988821199042

+ 1198971198881119904 + 119897

1198880

11990311988821199042 + 119903

1198881119904 + 120578

(43)









minus1 minus0944 + 119895004




minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048



selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)






7 Results


119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)



CK



0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)


minus4

minus2

0

2

4


0 50 100 150 200 250 300Time (s)

Curr

ent (

A)


minus05

0

05

1






119875) chosen from




0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





1198692= min

119870

(

1minus sum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816minus1003816100381610038161003816120596119905119903

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816120596119905119903

1003816100381610038161003816

forall10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816lt1003816100381610038161003816120596119905119903

1003816100381610038161003816

sum

119899

10038161003816100381610038161003816

1003816100381610038161003816I119898119862in1003816100381610038161003816 minus

10038161003816100381610038161003816I119898119862in

10038161003816100381610038161003816

10038161003816100381610038161003816

max (1003816100381610038161003816I119898119862in1003816100381610038161003816 I119898

10038161003816100381610038161003816119862in

10038161003816100381610038161003816)

) 119862in isin 119863in (8)






1198693= min

119870

(1 minussum

119899

10038161003816100381610038161003816

10038161003816100381610038161003816120593 119862in

10038161003816100381610038161003816minus10038161003816100381610038161205931205891003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161205931205891003816100381610038161003816

forall10038161003816100381610038161003816120593 119862in

10038161003816100381610038161003816lt10038161003816100381610038161205931205891003816100381610038161003816 )

119862in isin 119863in

(9)












(10)




119897 120596ℎ] 119888(119861) and 119889in(119861) ≫ 1 Under


119870(119904) = 119877

1015840







lim120596rarr119861

|119870 (119861)| ≫ 1 119861 = 119861 isin R | 0 lt 119861 le 120596low (11)


lim120596rarr119861

1003816100381610038161003816119878in (120575 120596)1003816100381610038161003816 = lim

120596rarr119861

(1003816100381610038161003816119878in (120596)

1003816100381610038161003816radic1205962 + 1205752)

lim120596rarr119861

|119878 (119861)| asymp 120575 997904rArr |119878 (120575 119861)| asymp 0

(12)


119877119870(119904) =

1198771015840

119870(119904)

119901

prod

119896=1

(119904 + 120575119896)

1198771015840

119870(119904) 119877

119870par (119904)

(13)




1198600(119904) 119877

1015840

119870(119904)

119901

prod

119896=1

(119904 + 120575119896) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

1198600(119904) 119877

119870(119904 120575) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

(14)



1198600(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119870(119904) 119877

119862(119904)

+ 1198610(119904) 119871

119870(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119862(119904)

+ 1198610(119904) 119861

1015840

(119904) 119871119862(119904) 119860

119898(119904) 119877

119870(119904)

+ 1198610(119904) 119861

1015840

(119904) 119861119898(119904) 119871

119870(119904) 119871

119862(119904) = 119862out (119904)

(15)


Γ (119904) 1198771015840

119862(119904)

119901

prod

119896=1

(119904 + 120578119896) + Υ (119904) 119871

119862(119904) = 119862out (119904)

Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871

119862(119904) = 119862out (119904)

(16)


Γ (119904) = 1198601015840

(119904) 119860119898(119904) (119860

0(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904))

Υ (119904) = 1198610(119904) 119861

1015840

(119904) (119860119898(119904) 119877

119870(119904) + 119861

119898(119904) 119871

119870(119904))

(17)







(18)


1ndash1198693



120590

p1

p3

p2

p2

p3

p1

j120596


minusj120596tr


119905119903

120590

p1

p3

p2

p2

p3

p1

j120596Damping line

120593120577

minus120593120577

120577

120593p1

120593p2





0isin





σ

Stable area

Intersection

120590ts low120590ts high j120596

p2

p1

p3

120593120577

120577

j120596tr

minusj120596tr

Ψ


c iPm(s) K(s 120590)

K(s 120590)

din

ΔP(s)y

120585998400w998400




119872= 119875

0(1 +Δ119882


0 The closed-loop


119872is

(

119910

) = (1 + 119870119875

0(1 + Δ119882

119872))minus1

times [1198750(1 + Δ119882

119872) (1 + 119870119875

119898) 119875

0(1 + Δ119882

119872)

1 + 119870119875119898

1](

119888

119889in)

(19)


10038171003817100381710038171003817100381710038171003817

Δ119882119872

1198701198750

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1


1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin

lt 1

(20)


+e c y

z

Ws(s)

C(s 120578) TP119898 in(s)y

P998400(s)

++minus

w1

dout

Vout(s)

120585Vnoise(s)

w2



=

1198750(1 + Δ119882


(

119910

) = (1 + 119870119875

0(1 + Δ119882

119868)minus1

)minus1

times [1198750(1 + Δ119882

119868)minus1

(1 + 119870119875119898) 119875

0(1 + Δ119882

119868)minus1

1 + 119870119875119898

1]

times (119888

119889in)

(21)


10038171003817100381710038171003817100381710038171003817

Δ119882119868

1

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1




1198791199081015840

1

= (1 + 119870 (120575) Δ119875)minus1

1198811015840

noise (23)


1198791199081015840

1

= min119870

10038171003817100381710038171003817100381710038171003817

[119882119872

119882119868] [119879in 0

0 119878in]119881

1015840

noise

10038171003817100381710038171003817100381710038171003817

(24)



infin






119904 119881out 119881noise is

119879119911119908= 119882

minus1

119904[(1 + 119862 (120578) 119879in119875

1015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

] [

[

1


]

]

= 119882minus1

119904[119878out 119878out 119878out] [

[

1


]

]

(25)


min119862

10038171003817100381710038171198791199111199081003817100381710038171003817infin

= 120574min (26)


1 119908



|119882119904|






minus1



119860(1205962

) minus 119861 (1205962

) = 120587 (1205962

) gt 0 (27)


Φ(119895120596) = 1205742

119868 minus119861 (119895120596)

119860 (119895120596)

119861 (minus119895120596)

119860 (minus119895120596) (28)




2

)119861(119895120596)119861(minus119895120596) = 119861(120596

2

) holds true





infin


is119861119860

infin= sup

120596|119861(119895120596)119860

minus1





minus1





119872(1205962


120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908

119886119872(minus119904)

minus 1198610(119904) 119861

0(minus119904) 119871

119870(119904 120575) 119871

119870(minus119904 120575) 119908

119887119872(119904)

times 119908119887119872

(minus119904)

120587119872(1205962

120575) = 119862in119908119886119872 (1205962

) minus 1198610119871119870119908119887119872

(1205962

120575) gt 0

(29)


119872(119904) = 119908

119887119872(119904)119908

minus1

119886119872(119904)

Even polynomial 120587119868(1205962


119868(119904) =

119908119887119868(119904)119908

minus1

119886119868(119904) is

120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)

minus 1198600(119904) 119860

0(minus119904) 119877

119870(119904 120575) 119877

119870(minus119904 120575) 119908

119887119868(119904)

times 119908119887119868(minus119904)

120587119868(1205962

120575) = 119862in119908119886119868 (1205962

) minus 1198600119877119870119908119887119868(1205962

120575) gt 0

(30)


Even polynomial 1205871198791199081015840

1

(1205962


119872(119904) and 119882


1198811015840

noise(119904) = V1015840119887noise(119904)V

1015840minus1

119886noise(119904) is

1205871198791199081015840

1119872

(1205962

120575) = 119908119886119872119862inV

1015840

119886noise (1205962

)

minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596

2

120575)

1205871198791199081015840

1119878

(1205962

120575) = 119908119886119878119862inV

1015840

119886out (1205962

)

minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596

2

120575)

1205871198791199081015840

1

(1205962

120575) = 05 (1205871198791199081015840

1119872

(1205962

120575) + 1205871198791199081015840

1119878

(1205962

120575)) gt 0

(31)



1198791199081015840

1

(1205962

120575)


1198791199081015840

1

(1205962

120575)


1119872

(1205962

120575) and1205871198791199081015840

1119878

(1205962

120575)


119878(119904) = 119908

119887119878(119904)119908

minus1

119886119878(119904) 119881out(119904) =

V119887out(119904)V

minus1


minus1

119886noise(119904) is

1205871199081119911(1205962

120578) = 119908119886119878119862out (120596

2

) minus 11990811988711987811986001198601198981198601015840

119877119870119877119862(1205962

120578)

1205871199082119911(1205962

120578) = 119908119886119878119862outV119886out (120596

2

)

minus 11990811988711987811986001198601198981198601015840

119877119870119877119862Vbout (120596

2

120578)

120587119911119908(1205962

120578) = (1205871199081119911(1205962

120578)2

+ 1205871199082119911(1205962

120578)2

) gt 0

(32)


119911119908(1205962



R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0

R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)


120587119870SPR

(1205962

120590)

= 2 (119877119870(minus119895120596 120590) 119871

119870(119895120596) + 119877

119870(119895120596 120590) 119871

119870(minus119895120596)) gt 0

120587119862SPR

(1205962

120578)

= 2 (119877119862(minus119895120596 120578) 119871

119862(119895120596) + 119877

119862(119895120596 120578) 119871

119862(minus119895120596)) gt 0

(34)









deg119877119870= deg 119871

119870


deg119877119862= deg 119871

119862






min1205962or119883

119891 (1205962

119883) =

[[[[[[[[[[[[[[[[

[

1198691(119883) and 119869out 1 (119883)

1198692(119883) and 119869out 2 (119883)

1198693(119883) and 119869out 3 (119883)

120587119872(1205962

119883)

120587119868(1205962

119883)

1205871198791199081015840

1

(1205962

119883)

120587119911119908(1205962

119883)

120587119870SPR

(1205962

119883)

120587119862SPR

(1205962

119883)

]]]]]]]]]]]]]]]]

]

st 119891 (1205962

119883) gt 0

119883 isin 119875

(35)


119870 119877

119870 119871

119862 119877






0




minus3 kgm2


119896119875

0081NmA119877 79Ω119871 57mH

C

LK RK

K

LC RC



(deg119862in+deg119875119898+deg1198751015840








1198750(119904) =

120596 (119904)

119894 (119904)=

119896119875

119869119904 + 119861=

1306

119904 + 0667

1198751015840

(119904) =120593 (119904)

120596 (119904)=1

119904

(36)




minus3 kgm2


Δ119896119875

[06 divide 13] NmA



119904lt 14 s



119890120596| lt |119880limit|




Δ119875 (119904) =Δ119896

119875

Δ119869119904 + Δ119861 (37)



119882119872(119904) =

134 times 1199043

+ 1156 times 1199042

+ 032 times 119904 + 0062

1199043 + 157 times 1199042 + 048 times 119904 + 0096

119882119868(119904) =

065 times 1199042

+ 039 times 119904 + 0083

1199042 + 082 times 119904 + 017

1198811015840

noise (119904) =00023 times 119904 + 12 sdot 10

minus6

119904 + 9803

(38)




0(119904) The selected 119875

119898(119904) is

119875119898(119904) =

2892

119904 + 15 (39)



120590119905119904 low minus07

120590119905119904 high minus4

plusmn119895120596119905119903

plusmn11989510

plusmn120593120577

plusmn30∘


in a complex plain


120590119905119904 low minus035

120590119905119904 high minus9

plusmn119895120596119905119903

plusmn11989514

plusmn120593120577

plusmn74∘


119882119878(119904) =

21 times 1199042

+ 046 times 119904 + 0001

1199042 + 398 times 119904 + 099

119881out (119904) =0002 times 119904 + 00012

119904 + 00014

119881noise (119904) =29 sdot 10

minus3


119904 + 1001 sdot 103

(40)


119870(119904 1199031198960) =

1198971198961119904 + 119897

1198960

1199031198961119904 + 120575

(41)


minus4

minus10minus2




polynomial is

119862in (119904) = 1199042

+ 2 times 119904 + 1 (42)



119862 (119904 1199031198880) =

11989711988821199042

+ 1198971198881119904 + 119897

1198880

11990311988821199042 + 119903

1198881119904 + 120578

(43)









minus1 minus0944 + 119895004




minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048



selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)






7 Results


119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)



CK



0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)


minus4

minus2

0

2

4


0 50 100 150 200 250 300Time (s)

Curr

ent (

A)


minus05

0

05

1






119875) chosen from




0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1198600(119904) 119877

1015840

119870(119904)

119901

prod

119896=1

(119904 + 120575119896) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

1198600(119904) 119877

119870(119904 120575) + 119861

0(119904) 119871

119870(119904) = 119862in (119904)

(14)



1198600(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119870(119904) 119877

119862(119904)

+ 1198610(119904) 119871

119870(119904) 119860

1015840

(119904) 119860119898(119904) 119877

119862(119904)

+ 1198610(119904) 119861

1015840

(119904) 119871119862(119904) 119860

119898(119904) 119877

119870(119904)

+ 1198610(119904) 119861

1015840

(119904) 119861119898(119904) 119871

119870(119904) 119871

119862(119904) = 119862out (119904)

(15)


Γ (119904) 1198771015840

119862(119904)

119901

prod

119896=1

(119904 + 120578119896) + Υ (119904) 119871

119862(119904) = 119862out (119904)

Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871

119862(119904) = 119862out (119904)

(16)


Γ (119904) = 1198601015840

(119904) 119860119898(119904) (119860

0(119904) 119877

119870(119904) + 119861

0(119904) 119871

119870(119904))

Υ (119904) = 1198610(119904) 119861

1015840

(119904) (119860119898(119904) 119877

119870(119904) + 119861

119898(119904) 119871

119870(119904))

(17)







(18)


1ndash1198693



120590

p1

p3

p2

p2

p3

p1

j120596


minusj120596tr


119905119903

120590

p1

p3

p2

p2

p3

p1

j120596Damping line

120593120577

minus120593120577

120577

120593p1

120593p2





0isin





σ

Stable area

Intersection

120590ts low120590ts high j120596

p2

p1

p3

120593120577

120577

j120596tr

minusj120596tr

Ψ


c iPm(s) K(s 120590)

K(s 120590)

din

ΔP(s)y

120585998400w998400




119872= 119875

0(1 +Δ119882


0 The closed-loop


119872is

(

119910

) = (1 + 119870119875

0(1 + Δ119882

119872))minus1

times [1198750(1 + Δ119882

119872) (1 + 119870119875

119898) 119875

0(1 + Δ119882

119872)

1 + 119870119875119898

1](

119888

119889in)

(19)


10038171003817100381710038171003817100381710038171003817

Δ119882119872

1198701198750

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1


1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin

lt 1

(20)


+e c y

z

Ws(s)

C(s 120578) TP119898 in(s)y

P998400(s)

++minus

w1

dout

Vout(s)

120585Vnoise(s)

w2



=

1198750(1 + Δ119882


(

119910

) = (1 + 119870119875

0(1 + Δ119882

119868)minus1

)minus1

times [1198750(1 + Δ119882

119868)minus1

(1 + 119870119875119898) 119875

0(1 + Δ119882

119868)minus1

1 + 119870119875119898

1]

times (119888

119889in)

(21)


10038171003817100381710038171003817100381710038171003817

Δ119882119868

1

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1




1198791199081015840

1

= (1 + 119870 (120575) Δ119875)minus1

1198811015840

noise (23)


1198791199081015840

1

= min119870

10038171003817100381710038171003817100381710038171003817

[119882119872

119882119868] [119879in 0

0 119878in]119881

1015840

noise

10038171003817100381710038171003817100381710038171003817

(24)



infin






119904 119881out 119881noise is

119879119911119908= 119882

minus1

119904[(1 + 119862 (120578) 119879in119875

1015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

] [

[

1


]

]

= 119882minus1

119904[119878out 119878out 119878out] [

[

1


]

]

(25)


min119862

10038171003817100381710038171198791199111199081003817100381710038171003817infin

= 120574min (26)


1 119908



|119882119904|






minus1



119860(1205962

) minus 119861 (1205962

) = 120587 (1205962

) gt 0 (27)


Φ(119895120596) = 1205742

119868 minus119861 (119895120596)

119860 (119895120596)

119861 (minus119895120596)

119860 (minus119895120596) (28)




2

)119861(119895120596)119861(minus119895120596) = 119861(120596

2

) holds true





infin


is119861119860

infin= sup

120596|119861(119895120596)119860

minus1





minus1





119872(1205962


120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908

119886119872(minus119904)

minus 1198610(119904) 119861

0(minus119904) 119871

119870(119904 120575) 119871

119870(minus119904 120575) 119908

119887119872(119904)

times 119908119887119872

(minus119904)

120587119872(1205962

120575) = 119862in119908119886119872 (1205962

) minus 1198610119871119870119908119887119872

(1205962

120575) gt 0

(29)


119872(119904) = 119908

119887119872(119904)119908

minus1

119886119872(119904)

Even polynomial 120587119868(1205962


119868(119904) =

119908119887119868(119904)119908

minus1

119886119868(119904) is

120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)

minus 1198600(119904) 119860

0(minus119904) 119877

119870(119904 120575) 119877

119870(minus119904 120575) 119908

119887119868(119904)

times 119908119887119868(minus119904)

120587119868(1205962

120575) = 119862in119908119886119868 (1205962

) minus 1198600119877119870119908119887119868(1205962

120575) gt 0

(30)


Even polynomial 1205871198791199081015840

1

(1205962


119872(119904) and 119882


1198811015840

noise(119904) = V1015840119887noise(119904)V

1015840minus1

119886noise(119904) is

1205871198791199081015840

1119872

(1205962

120575) = 119908119886119872119862inV

1015840

119886noise (1205962

)

minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596

2

120575)

1205871198791199081015840

1119878

(1205962

120575) = 119908119886119878119862inV

1015840

119886out (1205962

)

minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596

2

120575)

1205871198791199081015840

1

(1205962

120575) = 05 (1205871198791199081015840

1119872

(1205962

120575) + 1205871198791199081015840

1119878

(1205962

120575)) gt 0

(31)



1198791199081015840

1

(1205962

120575)


1198791199081015840

1

(1205962

120575)


1119872

(1205962

120575) and1205871198791199081015840

1119878

(1205962

120575)


119878(119904) = 119908

119887119878(119904)119908

minus1

119886119878(119904) 119881out(119904) =

V119887out(119904)V

minus1


minus1

119886noise(119904) is

1205871199081119911(1205962

120578) = 119908119886119878119862out (120596

2

) minus 11990811988711987811986001198601198981198601015840

119877119870119877119862(1205962

120578)

1205871199082119911(1205962

120578) = 119908119886119878119862outV119886out (120596

2

)

minus 11990811988711987811986001198601198981198601015840

119877119870119877119862Vbout (120596

2

120578)

120587119911119908(1205962

120578) = (1205871199081119911(1205962

120578)2

+ 1205871199082119911(1205962

120578)2

) gt 0

(32)


119911119908(1205962



R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0

R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)


120587119870SPR

(1205962

120590)

= 2 (119877119870(minus119895120596 120590) 119871

119870(119895120596) + 119877

119870(119895120596 120590) 119871

119870(minus119895120596)) gt 0

120587119862SPR

(1205962

120578)

= 2 (119877119862(minus119895120596 120578) 119871

119862(119895120596) + 119877

119862(119895120596 120578) 119871

119862(minus119895120596)) gt 0

(34)









deg119877119870= deg 119871

119870


deg119877119862= deg 119871

119862






min1205962or119883

119891 (1205962

119883) =

[[[[[[[[[[[[[[[[

[

1198691(119883) and 119869out 1 (119883)

1198692(119883) and 119869out 2 (119883)

1198693(119883) and 119869out 3 (119883)

120587119872(1205962

119883)

120587119868(1205962

119883)

1205871198791199081015840

1

(1205962

119883)

120587119911119908(1205962

119883)

120587119870SPR

(1205962

119883)

120587119862SPR

(1205962

119883)

]]]]]]]]]]]]]]]]

]

st 119891 (1205962

119883) gt 0

119883 isin 119875

(35)


119870 119877

119870 119871

119862 119877






0




minus3 kgm2


119896119875

0081NmA119877 79Ω119871 57mH

C

LK RK

K

LC RC



(deg119862in+deg119875119898+deg1198751015840








1198750(119904) =

120596 (119904)

119894 (119904)=

119896119875

119869119904 + 119861=

1306

119904 + 0667

1198751015840

(119904) =120593 (119904)

120596 (119904)=1

119904

(36)




minus3 kgm2


Δ119896119875

[06 divide 13] NmA



119904lt 14 s



119890120596| lt |119880limit|




Δ119875 (119904) =Δ119896

119875

Δ119869119904 + Δ119861 (37)



119882119872(119904) =

134 times 1199043

+ 1156 times 1199042

+ 032 times 119904 + 0062

1199043 + 157 times 1199042 + 048 times 119904 + 0096

119882119868(119904) =

065 times 1199042

+ 039 times 119904 + 0083

1199042 + 082 times 119904 + 017

1198811015840

noise (119904) =00023 times 119904 + 12 sdot 10

minus6

119904 + 9803

(38)




0(119904) The selected 119875

119898(119904) is

119875119898(119904) =

2892

119904 + 15 (39)



120590119905119904 low minus07

120590119905119904 high minus4

plusmn119895120596119905119903

plusmn11989510

plusmn120593120577

plusmn30∘


in a complex plain


120590119905119904 low minus035

120590119905119904 high minus9

plusmn119895120596119905119903

plusmn11989514

plusmn120593120577

plusmn74∘


119882119878(119904) =

21 times 1199042

+ 046 times 119904 + 0001

1199042 + 398 times 119904 + 099

119881out (119904) =0002 times 119904 + 00012

119904 + 00014

119881noise (119904) =29 sdot 10

minus3


119904 + 1001 sdot 103

(40)


119870(119904 1199031198960) =

1198971198961119904 + 119897

1198960

1199031198961119904 + 120575

(41)


minus4

minus10minus2




polynomial is

119862in (119904) = 1199042

+ 2 times 119904 + 1 (42)



119862 (119904 1199031198880) =

11989711988821199042

+ 1198971198881119904 + 119897

1198880

11990311988821199042 + 119903

1198881119904 + 120578

(43)









minus1 minus0944 + 119895004




minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048



selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)






7 Results


119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)



CK



0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)


minus4

minus2

0

2

4


0 50 100 150 200 250 300Time (s)

Curr

ent (

A)


minus05

0

05

1






119875) chosen from




0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




σ

Stable area

Intersection

120590ts low120590ts high j120596

p2

p1

p3

120593120577

120577

j120596tr

minusj120596tr

Ψ


c iPm(s) K(s 120590)

K(s 120590)

din

ΔP(s)y

120585998400w998400




119872= 119875

0(1 +Δ119882


0 The closed-loop


119872is

(

119910

) = (1 + 119870119875

0(1 + Δ119882

119872))minus1

times [1198750(1 + Δ119882

119872) (1 + 119870119875

119898) 119875

0(1 + Δ119882

119872)

1 + 119870119875119898

1](

119888

119889in)

(19)


10038171003817100381710038171003817100381710038171003817

Δ119882119872

1198701198750

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1


1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin

lt 1

(20)


+e c y

z

Ws(s)

C(s 120578) TP119898 in(s)y

P998400(s)

++minus

w1

dout

Vout(s)

120585Vnoise(s)

w2



=

1198750(1 + Δ119882


(

119910

) = (1 + 119870119875

0(1 + Δ119882

119868)minus1

)minus1

times [1198750(1 + Δ119882

119868)minus1

(1 + 119870119875119898) 119875

0(1 + Δ119882

119868)minus1

1 + 119870119875119898

1]

times (119888

119889in)

(21)


10038171003817100381710038171003817100381710038171003817

Δ119882119868

1

1 + 1198701198750

10038171003817100381710038171003817100381710038171003817infin

lt 1




1198791199081015840

1

= (1 + 119870 (120575) Δ119875)minus1

1198811015840

noise (23)


1198791199081015840

1

= min119870

10038171003817100381710038171003817100381710038171003817

[119882119872

119882119868] [119879in 0

0 119878in]119881

1015840

noise

10038171003817100381710038171003817100381710038171003817

(24)



infin






119904 119881out 119881noise is

119879119911119908= 119882

minus1

119904[(1 + 119862 (120578) 119879in119875

1015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

] [

[

1


]

]

= 119882minus1

119904[119878out 119878out 119878out] [

[

1


]

]

(25)


min119862

10038171003817100381710038171198791199111199081003817100381710038171003817infin

= 120574min (26)


1 119908



|119882119904|






minus1



119860(1205962

) minus 119861 (1205962

) = 120587 (1205962

) gt 0 (27)


Φ(119895120596) = 1205742

119868 minus119861 (119895120596)

119860 (119895120596)

119861 (minus119895120596)

119860 (minus119895120596) (28)




2

)119861(119895120596)119861(minus119895120596) = 119861(120596

2

) holds true





infin


is119861119860

infin= sup

120596|119861(119895120596)119860

minus1





minus1





119872(1205962


120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908

119886119872(minus119904)

minus 1198610(119904) 119861

0(minus119904) 119871

119870(119904 120575) 119871

119870(minus119904 120575) 119908

119887119872(119904)

times 119908119887119872

(minus119904)

120587119872(1205962

120575) = 119862in119908119886119872 (1205962

) minus 1198610119871119870119908119887119872

(1205962

120575) gt 0

(29)


119872(119904) = 119908

119887119872(119904)119908

minus1

119886119872(119904)

Even polynomial 120587119868(1205962


119868(119904) =

119908119887119868(119904)119908

minus1

119886119868(119904) is

120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)

minus 1198600(119904) 119860

0(minus119904) 119877

119870(119904 120575) 119877

119870(minus119904 120575) 119908

119887119868(119904)

times 119908119887119868(minus119904)

120587119868(1205962

120575) = 119862in119908119886119868 (1205962

) minus 1198600119877119870119908119887119868(1205962

120575) gt 0

(30)


Even polynomial 1205871198791199081015840

1

(1205962


119872(119904) and 119882


1198811015840

noise(119904) = V1015840119887noise(119904)V

1015840minus1

119886noise(119904) is

1205871198791199081015840

1119872

(1205962

120575) = 119908119886119872119862inV

1015840

119886noise (1205962

)

minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596

2

120575)

1205871198791199081015840

1119878

(1205962

120575) = 119908119886119878119862inV

1015840

119886out (1205962

)

minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596

2

120575)

1205871198791199081015840

1

(1205962

120575) = 05 (1205871198791199081015840

1119872

(1205962

120575) + 1205871198791199081015840

1119878

(1205962

120575)) gt 0

(31)



1198791199081015840

1

(1205962

120575)


1198791199081015840

1

(1205962

120575)


1119872

(1205962

120575) and1205871198791199081015840

1119878

(1205962

120575)


119878(119904) = 119908

119887119878(119904)119908

minus1

119886119878(119904) 119881out(119904) =

V119887out(119904)V

minus1


minus1

119886noise(119904) is

1205871199081119911(1205962

120578) = 119908119886119878119862out (120596

2

) minus 11990811988711987811986001198601198981198601015840

119877119870119877119862(1205962

120578)

1205871199082119911(1205962

120578) = 119908119886119878119862outV119886out (120596

2

)

minus 11990811988711987811986001198601198981198601015840

119877119870119877119862Vbout (120596

2

120578)

120587119911119908(1205962

120578) = (1205871199081119911(1205962

120578)2

+ 1205871199082119911(1205962

120578)2

) gt 0

(32)


119911119908(1205962



R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0

R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)


120587119870SPR

(1205962

120590)

= 2 (119877119870(minus119895120596 120590) 119871

119870(119895120596) + 119877

119870(119895120596 120590) 119871

119870(minus119895120596)) gt 0

120587119862SPR

(1205962

120578)

= 2 (119877119862(minus119895120596 120578) 119871

119862(119895120596) + 119877

119862(119895120596 120578) 119871

119862(minus119895120596)) gt 0

(34)









deg119877119870= deg 119871

119870


deg119877119862= deg 119871

119862






min1205962or119883

119891 (1205962

119883) =

[[[[[[[[[[[[[[[[

[

1198691(119883) and 119869out 1 (119883)

1198692(119883) and 119869out 2 (119883)

1198693(119883) and 119869out 3 (119883)

120587119872(1205962

119883)

120587119868(1205962

119883)

1205871198791199081015840

1

(1205962

119883)

120587119911119908(1205962

119883)

120587119870SPR

(1205962

119883)

120587119862SPR

(1205962

119883)

]]]]]]]]]]]]]]]]

]

st 119891 (1205962

119883) gt 0

119883 isin 119875

(35)


119870 119877

119870 119871

119862 119877






0




minus3 kgm2


119896119875

0081NmA119877 79Ω119871 57mH

C

LK RK

K

LC RC



(deg119862in+deg119875119898+deg1198751015840








1198750(119904) =

120596 (119904)

119894 (119904)=

119896119875

119869119904 + 119861=

1306

119904 + 0667

1198751015840

(119904) =120593 (119904)

120596 (119904)=1

119904

(36)




minus3 kgm2


Δ119896119875

[06 divide 13] NmA



119904lt 14 s



119890120596| lt |119880limit|




Δ119875 (119904) =Δ119896

119875

Δ119869119904 + Δ119861 (37)



119882119872(119904) =

134 times 1199043

+ 1156 times 1199042

+ 032 times 119904 + 0062

1199043 + 157 times 1199042 + 048 times 119904 + 0096

119882119868(119904) =

065 times 1199042

+ 039 times 119904 + 0083

1199042 + 082 times 119904 + 017

1198811015840

noise (119904) =00023 times 119904 + 12 sdot 10

minus6

119904 + 9803

(38)




0(119904) The selected 119875

119898(119904) is

119875119898(119904) =

2892

119904 + 15 (39)



120590119905119904 low minus07

120590119905119904 high minus4

plusmn119895120596119905119903

plusmn11989510

plusmn120593120577

plusmn30∘


in a complex plain


120590119905119904 low minus035

120590119905119904 high minus9

plusmn119895120596119905119903

plusmn11989514

plusmn120593120577

plusmn74∘


119882119878(119904) =

21 times 1199042

+ 046 times 119904 + 0001

1199042 + 398 times 119904 + 099

119881out (119904) =0002 times 119904 + 00012

119904 + 00014

119881noise (119904) =29 sdot 10

minus3


119904 + 1001 sdot 103

(40)


119870(119904 1199031198960) =

1198971198961119904 + 119897

1198960

1199031198961119904 + 120575

(41)


minus4

minus10minus2




polynomial is

119862in (119904) = 1199042

+ 2 times 119904 + 1 (42)



119862 (119904 1199031198880) =

11989711988821199042

+ 1198971198881119904 + 119897

1198880

11990311988821199042 + 119903

1198881119904 + 120578

(43)









minus1 minus0944 + 119895004




minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048



selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)






7 Results


119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)



CK



0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)


minus4

minus2

0

2

4


0 50 100 150 200 250 300Time (s)

Curr

ent (

A)


minus05

0

05

1






119875) chosen from




0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119904 119881out 119881noise is

119879119911119908= 119882

minus1

119904[(1 + 119862 (120578) 119879in119875

1015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

(1 + 119862 (120578) 119879in1198751015840

)minus1

] [

[

1


]

]

= 119882minus1

119904[119878out 119878out 119878out] [

[

1


]

]

(25)


min119862

10038171003817100381710038171198791199111199081003817100381710038171003817infin

= 120574min (26)


1 119908



|119882119904|






minus1



119860(1205962

) minus 119861 (1205962

) = 120587 (1205962

) gt 0 (27)


Φ(119895120596) = 1205742

119868 minus119861 (119895120596)

119860 (119895120596)

119861 (minus119895120596)

119860 (minus119895120596) (28)




2

)119861(119895120596)119861(minus119895120596) = 119861(120596

2

) holds true





infin


is119861119860

infin= sup

120596|119861(119895120596)119860

minus1





minus1





119872(1205962


120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908

119886119872(minus119904)

minus 1198610(119904) 119861

0(minus119904) 119871

119870(119904 120575) 119871

119870(minus119904 120575) 119908

119887119872(119904)

times 119908119887119872

(minus119904)

120587119872(1205962

120575) = 119862in119908119886119872 (1205962

) minus 1198610119871119870119908119887119872

(1205962

120575) gt 0

(29)


119872(119904) = 119908

119887119872(119904)119908

minus1

119886119872(119904)

Even polynomial 120587119868(1205962


119868(119904) =

119908119887119868(119904)119908

minus1

119886119868(119904) is

120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)

minus 1198600(119904) 119860

0(minus119904) 119877

119870(119904 120575) 119877

119870(minus119904 120575) 119908

119887119868(119904)

times 119908119887119868(minus119904)

120587119868(1205962

120575) = 119862in119908119886119868 (1205962

) minus 1198600119877119870119908119887119868(1205962

120575) gt 0

(30)


Even polynomial 1205871198791199081015840

1

(1205962


119872(119904) and 119882


1198811015840

noise(119904) = V1015840119887noise(119904)V

1015840minus1

119886noise(119904) is

1205871198791199081015840

1119872

(1205962

120575) = 119908119886119872119862inV

1015840

119886noise (1205962

)

minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596

2

120575)

1205871198791199081015840

1119878

(1205962

120575) = 119908119886119878119862inV

1015840

119886out (1205962

)

minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596

2

120575)

1205871198791199081015840

1

(1205962

120575) = 05 (1205871198791199081015840

1119872

(1205962

120575) + 1205871198791199081015840

1119878

(1205962

120575)) gt 0

(31)



1198791199081015840

1

(1205962

120575)


1198791199081015840

1

(1205962

120575)


1119872

(1205962

120575) and1205871198791199081015840

1119878

(1205962

120575)


119878(119904) = 119908

119887119878(119904)119908

minus1

119886119878(119904) 119881out(119904) =

V119887out(119904)V

minus1


minus1

119886noise(119904) is

1205871199081119911(1205962

120578) = 119908119886119878119862out (120596

2

) minus 11990811988711987811986001198601198981198601015840

119877119870119877119862(1205962

120578)

1205871199082119911(1205962

120578) = 119908119886119878119862outV119886out (120596

2

)

minus 11990811988711987811986001198601198981198601015840

119877119870119877119862Vbout (120596

2

120578)

120587119911119908(1205962

120578) = (1205871199081119911(1205962

120578)2

+ 1205871199082119911(1205962

120578)2

) gt 0

(32)


119911119908(1205962



R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0

R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)


120587119870SPR

(1205962

120590)

= 2 (119877119870(minus119895120596 120590) 119871

119870(119895120596) + 119877

119870(119895120596 120590) 119871

119870(minus119895120596)) gt 0

120587119862SPR

(1205962

120578)

= 2 (119877119862(minus119895120596 120578) 119871

119862(119895120596) + 119877

119862(119895120596 120578) 119871

119862(minus119895120596)) gt 0

(34)









deg119877119870= deg 119871

119870


deg119877119862= deg 119871

119862






min1205962or119883

119891 (1205962

119883) =

[[[[[[[[[[[[[[[[

[

1198691(119883) and 119869out 1 (119883)

1198692(119883) and 119869out 2 (119883)

1198693(119883) and 119869out 3 (119883)

120587119872(1205962

119883)

120587119868(1205962

119883)

1205871198791199081015840

1

(1205962

119883)

120587119911119908(1205962

119883)

120587119870SPR

(1205962

119883)

120587119862SPR

(1205962

119883)

]]]]]]]]]]]]]]]]

]

st 119891 (1205962

119883) gt 0

119883 isin 119875

(35)


119870 119877

119870 119871

119862 119877






0




minus3 kgm2


119896119875

0081NmA119877 79Ω119871 57mH

C

LK RK

K

LC RC



(deg119862in+deg119875119898+deg1198751015840








1198750(119904) =

120596 (119904)

119894 (119904)=

119896119875

119869119904 + 119861=

1306

119904 + 0667

1198751015840

(119904) =120593 (119904)

120596 (119904)=1

119904

(36)




minus3 kgm2


Δ119896119875

[06 divide 13] NmA



119904lt 14 s



119890120596| lt |119880limit|




Δ119875 (119904) =Δ119896

119875

Δ119869119904 + Δ119861 (37)



119882119872(119904) =

134 times 1199043

+ 1156 times 1199042

+ 032 times 119904 + 0062

1199043 + 157 times 1199042 + 048 times 119904 + 0096

119882119868(119904) =

065 times 1199042

+ 039 times 119904 + 0083

1199042 + 082 times 119904 + 017

1198811015840

noise (119904) =00023 times 119904 + 12 sdot 10

minus6

119904 + 9803

(38)




0(119904) The selected 119875

119898(119904) is

119875119898(119904) =

2892

119904 + 15 (39)



120590119905119904 low minus07

120590119905119904 high minus4

plusmn119895120596119905119903

plusmn11989510

plusmn120593120577

plusmn30∘


in a complex plain


120590119905119904 low minus035

120590119905119904 high minus9

plusmn119895120596119905119903

plusmn11989514

plusmn120593120577

plusmn74∘


119882119878(119904) =

21 times 1199042

+ 046 times 119904 + 0001

1199042 + 398 times 119904 + 099

119881out (119904) =0002 times 119904 + 00012

119904 + 00014

119881noise (119904) =29 sdot 10

minus3


119904 + 1001 sdot 103

(40)


119870(119904 1199031198960) =

1198971198961119904 + 119897

1198960

1199031198961119904 + 120575

(41)


minus4

minus10minus2




polynomial is

119862in (119904) = 1199042

+ 2 times 119904 + 1 (42)



119862 (119904 1199031198880) =

11989711988821199042

+ 1198971198881119904 + 119897

1198880

11990311988821199042 + 119903

1198881119904 + 120578

(43)









minus1 minus0944 + 119895004




minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048



selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)






7 Results


119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)



CK



0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)


minus4

minus2

0

2

4


0 50 100 150 200 250 300Time (s)

Curr

ent (

A)


minus05

0

05

1






119875) chosen from




0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Even polynomial 1205871198791199081015840

1

(1205962


119872(119904) and 119882


1198811015840

noise(119904) = V1015840119887noise(119904)V

1015840minus1

119886noise(119904) is

1205871198791199081015840

1119872

(1205962

120575) = 119908119886119872119862inV

1015840

119886noise (1205962

)

minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596

2

120575)

1205871198791199081015840

1119878

(1205962

120575) = 119908119886119878119862inV

1015840

119886out (1205962

)

minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596

2

120575)

1205871198791199081015840

1

(1205962

120575) = 05 (1205871198791199081015840

1119872

(1205962

120575) + 1205871198791199081015840

1119878

(1205962

120575)) gt 0

(31)



1198791199081015840

1

(1205962

120575)


1198791199081015840

1

(1205962

120575)


1119872

(1205962

120575) and1205871198791199081015840

1119878

(1205962

120575)


119878(119904) = 119908

119887119878(119904)119908

minus1

119886119878(119904) 119881out(119904) =

V119887out(119904)V

minus1


minus1

119886noise(119904) is

1205871199081119911(1205962

120578) = 119908119886119878119862out (120596

2

) minus 11990811988711987811986001198601198981198601015840

119877119870119877119862(1205962

120578)

1205871199082119911(1205962

120578) = 119908119886119878119862outV119886out (120596

2

)

minus 11990811988711987811986001198601198981198601015840

119877119870119877119862Vbout (120596

2

120578)

120587119911119908(1205962

120578) = (1205871199081119911(1205962

120578)2

+ 1205871199082119911(1205962

120578)2

) gt 0

(32)


119911119908(1205962



R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0

R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)


120587119870SPR

(1205962

120590)

= 2 (119877119870(minus119895120596 120590) 119871

119870(119895120596) + 119877

119870(119895120596 120590) 119871

119870(minus119895120596)) gt 0

120587119862SPR

(1205962

120578)

= 2 (119877119862(minus119895120596 120578) 119871

119862(119895120596) + 119877

119862(119895120596 120578) 119871

119862(minus119895120596)) gt 0

(34)









deg119877119870= deg 119871

119870


deg119877119862= deg 119871

119862






min1205962or119883

119891 (1205962

119883) =

[[[[[[[[[[[[[[[[

[

1198691(119883) and 119869out 1 (119883)

1198692(119883) and 119869out 2 (119883)

1198693(119883) and 119869out 3 (119883)

120587119872(1205962

119883)

120587119868(1205962

119883)

1205871198791199081015840

1

(1205962

119883)

120587119911119908(1205962

119883)

120587119870SPR

(1205962

119883)

120587119862SPR

(1205962

119883)

]]]]]]]]]]]]]]]]

]

st 119891 (1205962

119883) gt 0

119883 isin 119875

(35)


119870 119877

119870 119871

119862 119877






0




minus3 kgm2


119896119875

0081NmA119877 79Ω119871 57mH

C

LK RK

K

LC RC



(deg119862in+deg119875119898+deg1198751015840








1198750(119904) =

120596 (119904)

119894 (119904)=

119896119875

119869119904 + 119861=

1306

119904 + 0667

1198751015840

(119904) =120593 (119904)

120596 (119904)=1

119904

(36)




minus3 kgm2


Δ119896119875

[06 divide 13] NmA



119904lt 14 s



119890120596| lt |119880limit|




Δ119875 (119904) =Δ119896

119875

Δ119869119904 + Δ119861 (37)



119882119872(119904) =

134 times 1199043

+ 1156 times 1199042

+ 032 times 119904 + 0062

1199043 + 157 times 1199042 + 048 times 119904 + 0096

119882119868(119904) =

065 times 1199042

+ 039 times 119904 + 0083

1199042 + 082 times 119904 + 017

1198811015840

noise (119904) =00023 times 119904 + 12 sdot 10

minus6

119904 + 9803

(38)




0(119904) The selected 119875

119898(119904) is

119875119898(119904) =

2892

119904 + 15 (39)



120590119905119904 low minus07

120590119905119904 high minus4

plusmn119895120596119905119903

plusmn11989510

plusmn120593120577

plusmn30∘


in a complex plain


120590119905119904 low minus035

120590119905119904 high minus9

plusmn119895120596119905119903

plusmn11989514

plusmn120593120577

plusmn74∘


119882119878(119904) =

21 times 1199042

+ 046 times 119904 + 0001

1199042 + 398 times 119904 + 099

119881out (119904) =0002 times 119904 + 00012

119904 + 00014

119881noise (119904) =29 sdot 10

minus3


119904 + 1001 sdot 103

(40)


119870(119904 1199031198960) =

1198971198961119904 + 119897

1198960

1199031198961119904 + 120575

(41)


minus4

minus10minus2




polynomial is

119862in (119904) = 1199042

+ 2 times 119904 + 1 (42)



119862 (119904 1199031198880) =

11989711988821199042

+ 1198971198881119904 + 119897

1198880

11990311988821199042 + 119903

1198881119904 + 120578

(43)









minus1 minus0944 + 119895004




minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048



selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)






7 Results


119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)



CK



0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)


minus4

minus2

0

2

4


0 50 100 150 200 250 300Time (s)

Curr

ent (

A)


minus05

0

05

1






119875) chosen from




0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






deg119877119870= deg 119871

119870


deg119877119862= deg 119871

119862






min1205962or119883

119891 (1205962

119883) =

[[[[[[[[[[[[[[[[

[

1198691(119883) and 119869out 1 (119883)

1198692(119883) and 119869out 2 (119883)

1198693(119883) and 119869out 3 (119883)

120587119872(1205962

119883)

120587119868(1205962

119883)

1205871198791199081015840

1

(1205962

119883)

120587119911119908(1205962

119883)

120587119870SPR

(1205962

119883)

120587119862SPR

(1205962

119883)

]]]]]]]]]]]]]]]]

]

st 119891 (1205962

119883) gt 0

119883 isin 119875

(35)


119870 119877

119870 119871

119862 119877






0




minus3 kgm2


119896119875

0081NmA119877 79Ω119871 57mH

C

LK RK

K

LC RC



(deg119862in+deg119875119898+deg1198751015840








1198750(119904) =

120596 (119904)

119894 (119904)=

119896119875

119869119904 + 119861=

1306

119904 + 0667

1198751015840

(119904) =120593 (119904)

120596 (119904)=1

119904

(36)




minus3 kgm2


Δ119896119875

[06 divide 13] NmA



119904lt 14 s



119890120596| lt |119880limit|




Δ119875 (119904) =Δ119896

119875

Δ119869119904 + Δ119861 (37)



119882119872(119904) =

134 times 1199043

+ 1156 times 1199042

+ 032 times 119904 + 0062

1199043 + 157 times 1199042 + 048 times 119904 + 0096

119882119868(119904) =

065 times 1199042

+ 039 times 119904 + 0083

1199042 + 082 times 119904 + 017

1198811015840

noise (119904) =00023 times 119904 + 12 sdot 10

minus6

119904 + 9803

(38)




0(119904) The selected 119875

119898(119904) is

119875119898(119904) =

2892

119904 + 15 (39)



120590119905119904 low minus07

120590119905119904 high minus4

plusmn119895120596119905119903

plusmn11989510

plusmn120593120577

plusmn30∘


in a complex plain


120590119905119904 low minus035

120590119905119904 high minus9

plusmn119895120596119905119903

plusmn11989514

plusmn120593120577

plusmn74∘


119882119878(119904) =

21 times 1199042

+ 046 times 119904 + 0001

1199042 + 398 times 119904 + 099

119881out (119904) =0002 times 119904 + 00012

119904 + 00014

119881noise (119904) =29 sdot 10

minus3


119904 + 1001 sdot 103

(40)


119870(119904 1199031198960) =

1198971198961119904 + 119897

1198960

1199031198961119904 + 120575

(41)


minus4

minus10minus2




polynomial is

119862in (119904) = 1199042

+ 2 times 119904 + 1 (42)



119862 (119904 1199031198880) =

11989711988821199042

+ 1198971198881119904 + 119897

1198880

11990311988821199042 + 119903

1198881119904 + 120578

(43)









minus1 minus0944 + 119895004




minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048



selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)






7 Results


119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)



CK



0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)


minus4

minus2

0

2

4


0 50 100 150 200 250 300Time (s)

Curr

ent (

A)


minus05

0

05

1






119875) chosen from




0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






minus3 kgm2


Δ119896119875

[06 divide 13] NmA



119904lt 14 s



119890120596| lt |119880limit|




Δ119875 (119904) =Δ119896

119875

Δ119869119904 + Δ119861 (37)



119882119872(119904) =

134 times 1199043

+ 1156 times 1199042

+ 032 times 119904 + 0062

1199043 + 157 times 1199042 + 048 times 119904 + 0096

119882119868(119904) =

065 times 1199042

+ 039 times 119904 + 0083

1199042 + 082 times 119904 + 017

1198811015840

noise (119904) =00023 times 119904 + 12 sdot 10

minus6

119904 + 9803

(38)




0(119904) The selected 119875

119898(119904) is

119875119898(119904) =

2892

119904 + 15 (39)



120590119905119904 low minus07

120590119905119904 high minus4

plusmn119895120596119905119903

plusmn11989510

plusmn120593120577

plusmn30∘


in a complex plain


120590119905119904 low minus035

120590119905119904 high minus9

plusmn119895120596119905119903

plusmn11989514

plusmn120593120577

plusmn74∘


119882119878(119904) =

21 times 1199042

+ 046 times 119904 + 0001

1199042 + 398 times 119904 + 099

119881out (119904) =0002 times 119904 + 00012

119904 + 00014

119881noise (119904) =29 sdot 10

minus3


119904 + 1001 sdot 103

(40)


119870(119904 1199031198960) =

1198971198961119904 + 119897

1198960

1199031198961119904 + 120575

(41)


minus4

minus10minus2




polynomial is

119862in (119904) = 1199042

+ 2 times 119904 + 1 (42)



119862 (119904 1199031198880) =

11989711988821199042

+ 1198971198881119904 + 119897

1198880

11990311988821199042 + 119903

1198881119904 + 120578

(43)









minus1 minus0944 + 119895004




minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048



selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)






7 Results


119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)



CK



0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)


minus4

minus2

0

2

4


0 50 100 150 200 250 300Time (s)

Curr

ent (

A)


minus05

0

05

1






119875) chosen from




0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in











minus1 minus0944 + 119895004




minus34 + 119895123 minus309 + 119895102


minus25 + 1198953 minus261 + 119895276

minus12 minus109

minus05 minus048



selected as

119862out (119904) = 1199046

+ 135 times 1199045

+ 234 times 1199044

+ 1286 times 1199043

+ 4171 times 1199042

+ 4773 times 119904 + 1490

(44)






7 Results


119870 (119904) =1356 sdot 10

minus3


119904 + 018 sdot 10minus3

119862 (119904) =027 times 119904

2

+ 23 times 119904 + 229

1199042 + 9324 times 119904 + 1021

(45)



CK



0 50 100 150 200 250 300Time (s)

Ang

ular

pos

ition

(rad

)


minus4

minus2

0

2

4


0 50 100 150 200 250 300Time (s)

Curr

ent (

A)


minus05

0

05

1






119875) chosen from




0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





0 50 100 150 200 250 300Time (s)

Ang

le er

ror (

rad)

minus05

0

05



0 50 100 150 200 250 300Time (s)

Volta

ge (V

)

minus20

minus10

0

10

20




1199081015840

1

infin



8 Conclusion


0 20 40 60 80 100Time (s)

Ang

ular

pos

ition

(rad

)

Output disturbance

Reference


minus1

0

1

2

3




0 20 40 60 80 100Time (s)

Curr

ent (

A)

minus02

minus01

0

01

02

03



Criteria sdot infin

Value1003817100381710038171003817119879in119882119872

1003817100381710038171003817infin082

1003817100381710038171003817119878in119882119868

1003817100381710038171003817infin062

1003817100381710038171003817119879119908101584011003817100381710038171003817infin

023

1003817100381710038171003817119878out119882minus1

119904

1003817100381710038171003817infin073

1003817100381710038171003817119882minus1

119904119878out119882out

1003817100381710038171003817infin0123

1003817100381710038171003817119882minus1

119904119878out119882noise

1003817100381710038171003817infin047






0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





0 20 40 60 80 100Time (s)

Ang

le er

ror (

rad)

minus02

minus01

0

01

02

03



0 20 40 60 80 100Time (s)

Volta

ge (V

)

minus2

0

2

4

6





References



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in

































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



research article optimal robust motion controller design

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